Integrand size = 33, antiderivative size = 601 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^{3/2} \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
1/4*b^(3/2)*(63*A*a^4*b+46*A*a^2*b^3+15*A*b^5-35*B*a^5-6*B*a^3*b^2-3*B*a*b ^4)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(7/2)/(a^2+b^2)^3/d+1/2*b*( A*b-B*a)*cot(d*x+c)^(5/2)/a/(a^2+b^2)/d/(b+a*cot(d*x+c))^2+1/4*b*(13*A*a^2 *b+5*A*b^3-9*B*a^3-B*a*b^2)*cot(d*x+c)^(3/2)/a^2/(a^2+b^2)^2/d/(b+a*cot(d* x+c))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(-1+2^(1 /2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3 *a^2*b*(A+B)-b^3*(A+B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2 ^(1/2)+1/4*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*ln(1+cot(d*x+ c)-2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(3*a^2*b*(A-B)-b^3* (A-B)-a^3*(A+B)+3*a*b^2*(A+B))*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/( a^2+b^2)^3/d*2^(1/2)-1/4*(8*A*a^4+31*A*a^2*b^2+15*A*b^4-11*B*a^3*b-3*B*a*b ^3)*cot(d*x+c)^(1/2)/a^3/(a^2+b^2)^2/d
Time = 6.56 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{4 \left (a^2+b^2\right )^3}-\frac {3 b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{7/2} \left (a^2+b^2\right )}-\frac {b^{3/2} \left (3 a^2 A b+A b^3-2 a^3 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{7/2} \left (a^2+b^2\right )^2}-\frac {b^{3/2} \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \left (\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}-\frac {A}{a^3 \sqrt {\tan (c+d x)}}-\frac {b^2 (A b-a B) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {3 b^2 (A b-a B) \sqrt {\tan (c+d x)}}{8 a^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {b^2 \left (3 a^2 A b+A b^3-2 a^3 B\right ) \sqrt {\tan (c+d x)}}{2 a^3 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{d} \]
(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3 ) - (3*b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/( 8*a^(7/2)*(a^2 + b^2)) - (b^(3/2)*(3*a^2*A*b + A*b^3 - 2*a^3*B)*ArcTan[(Sq rt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(7/2)*(a^2 + b^2)^2) - (b^(3/2)*( 6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B)*ArcTan[(Sqrt[b]*Sqr t[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(a^2 + b^2)^3) + ((3*a^2*b*(A - B) - b ^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[ Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) - A/(a^3*Sqrt[Tan[c + d*x]]) - (b^2*( A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (3*b^2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(8*a^3*(a^2 + b^2)*(a + b*Tan[c + d*x])) - (b^2*(3*a^2*A*b + A*b^3 - 2*a^3*B)*Sqrt[Tan[c + d*x]])/(2*a^3*(a ^2 + b^2)^2*(a + b*Tan[c + d*x]))))/d
Time = 3.15 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.87, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.879, Rules used = {3042, 4064, 3042, 4088, 27, 3042, 4128, 27, 3042, 4130, 27, 3042, 4136, 27, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{3/2} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {7}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {\cot ^{\frac {3}{2}}(c+d x) \left (\left (4 A a^2-b B a+5 A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+5 b (A b-a B)\right )}{2 (b+a \cot (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (\left (4 A a^2-b B a+5 A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+5 b (A b-a B)\right )}{(b+a \cot (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\left (4 A a^2-b B a+5 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+4 a (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )+5 b (A b-a B)\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {\sqrt {\cot (c+d x)} \left (-8 \left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2+\left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \cot ^2(c+d x)+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right )\right )}{2 (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\cot (c+d x)} \left (-8 \left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2+\left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \cot ^2(c+d x)+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right )\right )}{b+a \cot (c+d x)}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (8 \left (-B a^2+2 A b a+b^2 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right )\right )}{b-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {8 \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x) a^3+\left (-8 B a^5+24 A b a^4-3 b^2 B a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5\right ) \cot ^2(c+d x)+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {8 \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x) a^3+\left (-8 B a^5+24 A b a^4-3 b^2 B a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5\right ) \cot ^2(c+d x)+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-8 \left (A a^2+2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (-8 B a^5+24 A b a^4-3 b^2 B a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {8 \left (\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) a^3+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x) a^3\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {8 \int \frac {\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) a^3+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x) a^3}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {8 \int \frac {a^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-a^3 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 \int -\frac {a^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int \frac {a^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \int \frac {A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {2 b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right ) \sqrt {\cot (c+d x)}}{a d}-\frac {\frac {2 b^{3/2} \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b*(A*b - a*B)*Cot[c + d*x]^(5/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x]) ^2) + ((b*(13*a^2*A*b + 5*A*b^3 - 9*a^3*B - a*b^2*B)*Cot[c + d*x]^(3/2))/( a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((-2*(8*a^4*A + 31*a^2*A*b^2 + 15* A*b^4 - 11*a^3*b*B - 3*a*b^3*B)*Sqrt[Cot[c + d*x]])/(a*d) - ((2*b^(3/2)*(6 3*a^4*A*b + 46*a^2*A*b^3 + 15*A*b^5 - 35*a^5*B - 6*a^3*b^2*B - 3*a*b^4*B)* ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - (16*a^3* (((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(-(ArcTa n[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3* a*b^2*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sq rt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2 ))/((a^2 + b^2)*d))/a)/(2*a*(a^2 + b^2)))/(4*a*(a^2 + b^2))
3.6.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x ])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] & & LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(4580\) vs. \(2(549)=1098\).
Time = 0.42 (sec) , antiderivative size = 4581, normalized size of antiderivative = 7.62
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(4581\) |
| default | \(\text {Expression too large to display}\) | \(4581\) |
-1/4/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(15*A*arctan(b*tan(d*x+c)^(1/2)/(a* b)^(1/2))*tan(d*x+c)^(5/2)*b^9+15*A*(a*b)^(1/2)*tan(d*x+c)^2*b^8+24*A*(a*b )^(1/2)*a^6*b^2+24*A*(a*b)^(1/2)*a^4*b^4+8*A*(a*b)^(1/2)*a^2*b^6+8*A*(a*b) ^(1/2)*a^8+39*A*(a*b)^(1/2)*tan(d*x+c)^2*a^4*b^4+46*A*(a*b)^(1/2)*tan(d*x+ c)^2*a^2*b^6-11*B*(a*b)^(1/2)*tan(d*x+c)^2*a^5*b^3-14*B*(a*b)^(1/2)*tan(d* x+c)^2*a^3*b^5-3*B*(a*b)^(1/2)*tan(d*x+c)^2*a*b^7+65*A*(a*b)^(1/2)*tan(d*x +c)*a^5*b^3+74*A*(a*b)^(1/2)*tan(d*x+c)*a^3*b^5+25*A*(a*b)^(1/2)*tan(d*x+c )*a*b^7+8*A*(a*b)^(1/2)*a^6*b^2*tan(d*x+c)^2+63*A*arctan(b*tan(d*x+c)^(1/2 )/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a^6*b^3+46*A*arctan(b*tan(d*x+c)^(1/2)/(a* b)^(1/2))*tan(d*x+c)^(1/2)*a^4*b^5+15*A*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1 /2))*tan(d*x+c)^(1/2)*a^2*b^7-13*B*(a*b)^(1/2)*tan(d*x+c)*a^6*b^2-18*B*(a* b)^(1/2)*tan(d*x+c)*a^4*b^4-5*B*(a*b)^(1/2)*tan(d*x+c)*a^2*b^6-35*B*arctan (b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a^7*b^2-6*B*arctan(b*tan (d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a^5*b^4-3*B*arctan(b*tan(d*x+c )^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a^3*b^6+16*A*(a*b)^(1/2)*a^7*b*tan(d *x+c)-70*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a^6*b^3 -12*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a^4*b^5-6*B* arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a^2*b^7+63*A*arcta n(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(5/2)*a^4*b^5+46*A*arctan(b*t an(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(5/2)*a^2*b^7-35*B*arctan(b*tan...
Leaf count of result is larger than twice the leaf count of optimal. 8979 vs. \(2 (544) = 1088\).
Time = 187.83 (sec) , antiderivative size = 17988, normalized size of antiderivative = 29.93 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (35 \, B a^{5} b^{2} - 63 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 46 \, A a^{2} b^{5} + 3 \, B a b^{6} - 15 \, A b^{7}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {11 \, B a^{3} b^{3} - 15 \, A a^{2} b^{4} + 3 \, B a b^{5} - 7 \, A b^{6}}{\sqrt {\tan \left (d x + c\right )}} + \frac {13 \, B a^{4} b^{2} - 17 \, A a^{3} b^{3} + 5 \, B a^{2} b^{4} - 9 \, A a b^{5}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6} + \frac {2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}}{\tan \left (d x + c\right )^{2}}} + \frac {8 \, A}{a^{3} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]
-1/4*((35*B*a^5*b^2 - 63*A*a^4*b^3 + 6*B*a^3*b^4 - 46*A*a^2*b^5 + 3*B*a*b^ 6 - 15*A*b^7)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*sqrt(a*b)) - (2*sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2 *b - 3*(A - B)*a*b^2 - (A + B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(t an(d*x + c)))) + 2*sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^ 2 - (A + B)*b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + s qrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*log (sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*log(-sqrt(2)/sqrt(tan(d* x + c)) + 1/tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((11* B*a^3*b^3 - 15*A*a^2*b^4 + 3*B*a*b^5 - 7*A*b^6)/sqrt(tan(d*x + c)) + (13*B *a^4*b^2 - 17*A*a^3*b^3 + 5*B*a^2*b^4 - 9*A*a*b^5)/tan(d*x + c)^(3/2))/(a^ 7*b^2 + 2*a^5*b^4 + a^3*b^6 + 2*(a^8*b + 2*a^6*b^3 + a^4*b^5)/tan(d*x + c) + (a^9 + 2*a^7*b^2 + a^5*b^4)/tan(d*x + c)^2) + 8*A/(a^3*sqrt(tan(d*x + c ))))/d
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Hanged} \]