Integrand size = 33, antiderivative size = 534 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {b} \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
1/2*b*(A*b-B*a)*cot(d*x+c)^(3/2)/a/(a^2+b^2)/d/(b+a*cot(d*x+c))^2+1/2*(3*a ^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(-1+2^(1/2)*cot(d*x+c) ^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b ^2*(A+B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a^ 3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(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*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b* (A+B)-b^3*(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*(35*A*a^4*b+6*A*a^2*b^3+3*A*b^5-15*B*a^5+18*B*a^3*b^2+B*a*b^4)* arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))*b^(1/2)/a^(5/2)/(a^2+b^2)^3/d+1/4 *b*(11*A*a^2*b+3*A*b^3-7*B*a^3+B*a*b^2)*cot(d*x+c)^(1/2)/a^2/(a^2+b^2)^2/d /(b+a*cot(d*x+c))
Time = 6.47 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \left (\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{4 \left (a^2+b^2\right )^3}+\frac {3 \sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} \left (a^2+b^2\right )}+\frac {\sqrt {b} \left (2 a A b-a^2 B+b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{3/2} \left (a^2+b^2\right )^2}+\frac {\sqrt {b} \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^3}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {3 b (A b-a B) \sqrt {\tan (c+d x)}}{8 a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {b \left (2 a A b-a^2 B+b^2 B\right ) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{d} \]
(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(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*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/( 8*a^(5/2)*(a^2 + b^2)) + (Sqrt[b]*(2*a*A*b - a^2*B + b^2*B)*ArcTan[(Sqrt[b ]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(3/2)*(a^2 + b^2)^2) + (Sqrt[b]*(3*a^ 2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqr t[a]])/(Sqrt[a]*(a^2 + b^2)^3) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b *(A + B) - b^3*(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) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (3*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(8*a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])) + (b*(2*a*A*b - a^2*B + b^2*B)*Sqrt[Tan[c + d*x ]])/(2*a*(a^2 + b^2)^2*(a + b*Tan[c + d*x]))))/d
Time = 2.38 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.87, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.788, Rules used = {3042, 4064, 3042, 4088, 27, 3042, 4128, 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.
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {\sqrt {\cot (c+d x)} \left (\left (4 A a^2+b B a+3 A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+3 b (A b-a B)\right )}{2 (b+a \cot (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {\cot (c+d x)} \left (\left (4 A a^2+b B a+3 A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+3 b (A b-a B)\right )}{(b+a \cot (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (\left (4 A a^2+b B a+3 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+4 a (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )+3 b (A b-a B)\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2+\left (8 A a^4+9 b B a^3+3 A b^2 a^2+b^3 B a+3 A b^4\right ) \cot ^2(c+d x)+b \left (-7 B a^3+11 A b a^2+b^2 B a+3 A b^3\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2+\left (8 A a^4+9 b B a^3+3 A b^2 a^2+b^3 B a+3 A b^4\right ) \cot ^2(c+d x)+b \left (-7 B a^3+11 A b a^2+b^2 B a+3 A b^3\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 \left (-B a^2+2 A b a+b^2 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (8 A a^4+9 b B a^3+3 A b^2 a^2+b^3 B a+3 A b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-7 B a^3+11 A b a^2+b^2 B a+3 A b^3\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {8 \left (a^2 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right )-a^2 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a^2 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right )-a^2 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) a^2+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {a^2 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3-\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 \int \frac {a^2 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3-\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \int \frac {-B a^3+3 A b a^2+3 b^2 B a-A b^3-\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 a^2 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 a^2 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {\frac {16 a^2 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 \sqrt {b} \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b*(A*b - a*B)*Cot[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x]) ^2) + ((b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cot[c + d*x]])/( a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((2*Sqrt[b]*(35*a^4*A*b + 6*a^2*A* b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3*b^2*B + a*b^4*B)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a^2*(((3*a^2*b*(A - B) - b^ 3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[ c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(-1/2*Lo g[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2] *Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/(2* a*(a^2 + b^2)))/(4*a*(a^2 + b^2))
3.7.1.3.1 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[(((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. \(4190\) vs. \(2(486)=972\).
Time = 0.40 (sec) , antiderivative size = 4191, normalized size of antiderivative = 7.85
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(4191\) |
| default | \(\text {Expression too large to display}\) | \(4191\) |
1/4/d*(1/tan(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*(-15*B*arctan(b*tan(d*x+c)^(1/ 2)/(a*b)^(1/2))*a^5*b^3*tan(d*x+c)^2+18*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^ (1/2))*a^3*b^5*tan(d*x+c)^2+B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a*b^7 *tan(d*x+c)^2-7*B*(a*b)^(1/2)*tan(d*x+c)^(3/2)*a^5*b^2-6*B*(a*b)^(1/2)*tan (d*x+c)^(3/2)*a^3*b^4+B*(a*b)^(1/2)*tan(d*x+c)^(3/2)*a*b^6+70*A*arctan(b*t an(d*x+c)^(1/2)/(a*b)^(1/2))*a^5*b^3*tan(d*x+c)+12*A*arctan(b*tan(d*x+c)^( 1/2)/(a*b)^(1/2))*a^3*b^5*tan(d*x+c)+6*A*arctan(b*tan(d*x+c)^(1/2)/(a*b)^( 1/2))*a*b^7*tan(d*x+c)+A*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1 /2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*(a*b)^(1/2)*a^7+2*A*arctan(1+2 ^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^7+2*A*arctan(-1+2^(1/2)*tan (d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^7-30*B*arctan(b*tan(d*x+c)^(1/2)/(a*b )^(1/2))*a^6*b^2*tan(d*x+c)+36*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^ 4*b^4*tan(d*x+c)+2*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^2*b^6*tan(d* x+c)+2*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^7+2*B*ar ctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^7+B*ln(-(2^(1/2)*t an(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^( 1/2)*(a*b)^(1/2)*a^7+13*A*(a*b)^(1/2)*tan(d*x+c)^(1/2)*a^5*b^2+18*A*(a*b)^ (1/2)*tan(d*x+c)^(1/2)*a^3*b^4+5*A*(a*b)^(1/2)*tan(d*x+c)^(1/2)*a*b^6-9*B* (a*b)^(1/2)*tan(d*x+c)^(1/2)*a^6*b-2*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)) *2^(1/2)*(a*b)^(1/2)*a^4*b^3-3*B*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+...
Leaf count of result is larger than twice the leaf count of optimal. 8921 vs. \(2 (484) = 968\).
Time = 124.59 (sec) , antiderivative size = 17874, normalized size of antiderivative = 33.47 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\cot {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (15 \, B a^{5} b - 35 \, A a^{4} b^{2} - 18 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4} - B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} 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 {7 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3} - B a b^{4} - 3 \, A b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {9 \, B a^{4} b - 13 \, A a^{3} b^{2} + B a^{2} b^{3} - 5 \, A a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + \frac {2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]
1/4*((15*B*a^5*b - 35*A*a^4*b^2 - 18*B*a^3*b^3 - 6*A*a^2*b^4 - B*a*b^5 - 3 *A*b^6)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^8 + 3*a^6*b^2 + 3*a^4 *b^4 + a^2*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(tan(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)))) - 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) + 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) - ((7*B*a^3*b ^2 - 11*A*a^2*b^3 - B*a*b^4 - 3*A*b^5)/sqrt(tan(d*x + c)) + (9*B*a^4*b - 1 3*A*a^3*b^2 + B*a^2*b^3 - 5*A*a*b^4)/tan(d*x + c)^(3/2))/(a^6*b^2 + 2*a^4* b^4 + a^2*b^6 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)/tan(d*x + c) + (a^8 + 2*a^ 6*b^2 + a^4*b^4)/tan(d*x + c)^2))/d
\[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\cot \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]