Integrand size = 33, antiderivative size = 534 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 a^3 b^2 B-3 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{3/2} \sqrt {b} \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {\left (9 a^2 A b+A b^3-5 a^3 B+3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
-1/2*(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)*c ot(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*(15*A*a^4*b-18*A*a^2*b^3-A*b^5-3*B*a^5+26*B*a^3*b^2-3* B*a*b^4)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(3/2)/(a^2+b^2)^3/d/b^ (1/2)+1/2*b*(A*b-B*a)*cot(d*x+c)^(1/2)/a/(a^2+b^2)/d/(b+a*cot(d*x+c))^2-1/ 4*(9*A*a^2*b+A*b^3-5*B*a^3+3*B*a*b^2)*cot(d*x+c)^(1/2)/a/(a^2+b^2)^2/d/(b+ a*cot(d*x+c))
Time = 6.44 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{4 \left (a^2+b^2\right )^3}-\frac {3 (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} \sqrt {b} \left (a^2+b^2\right )}-\frac {\sqrt {b} \left (a^2 A-A b^2+2 a b B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{3/2} \left (a^2+b^2\right )^2}-\frac {\sqrt {b} \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \left (\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {3 (A b-a B) \sqrt {\tan (c+d x)}}{8 a \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {b \left (a^2 A-A b^2+2 a b B\right ) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{d} \]
(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*((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]]]))/(a^2 + b^2) ^3 - (3*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(8*a^(3/ 2)*Sqrt[b]*(a^2 + b^2)) - (Sqrt[b]*(a^2*A - A*b^2 + 2*a*b*B)*ArcTan[(Sqrt[ b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(3/2)*(a^2 + b^2)^2) - (Sqrt[b]*(a^3 *A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sq rt[a]])/(Sqrt[a]*(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*b - a*B)*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)*(a + b *Tan[c + d*x])^2) - (3*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(8*a*(a^2 + b^2)*(a + b*Tan[c + d*x])) - (b*(a^2*A - A*b^2 + 2*a*b*B)*Sqrt[Tan[c + d*x]])/(2* a*(a^2 + b^2)^2*(a + b*Tan[c + d*x]))))/d
Time = 2.36 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.86, 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, 4132, 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.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {\left (4 A a^2+3 b B a+A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+b (A b-a B)}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (4 A a^2+3 b B a+A b^2\right ) \cot ^2(c+d x)-4 a (A b-a B) \cot (c+d x)+b (A b-a B)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (4 A a^2+3 b B a+A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+4 a (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )+b (A b-a B)}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int \frac {-b \left (-5 B a^3+9 A b a^2+3 b^2 B a+A b^3\right ) \cot ^2(c+d x)-8 a b \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x)+b \left (-3 B a^3+7 A b a^2+5 b^2 B a-A b^3\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {-b \left (-5 B a^3+9 A b a^2+3 b^2 B a+A b^3\right ) \cot ^2(c+d x)-8 a b \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x)+b \left (-3 B a^3+7 A b a^2+5 b^2 B a-A b^3\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-b \left (-5 B a^3+9 A b a^2+3 b^2 B a+A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+8 a b \left (A a^2+2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b \left (-3 B a^3+7 A b a^2+5 b^2 B a-A b^3\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {8 \left (a b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )+a b \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )+a b \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {a b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-a b \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {a b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {16 \int \frac {a b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \int \frac {A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B) \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {-\frac {\left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\frac {16 a b \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 \sqrt {b} \left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x]) ^2) + (-(((9*a^2*A*b + A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cot[c + d*x]])/(( a^2 + b^2)*d*(b + a*Cot[c + d*x]))) - ((2*Sqrt[b]*(15*a^4*A*b - 18*a^2*A*b ^3 - A*b^5 - 3*a^5*B + 26*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*b*(((a^3*(A - B) - 3*a*b^2* (A - B) + 3*a^2*b*(A + B) - b^3*(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 - ( (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]]/Sqrt[2] + Log[1 + Sqrt[2]*S qrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/(2*b* (a^2 + b^2)))/(4*a*(a^2 + b^2))
3.7.2.3.1 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*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !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]
Time = 0.51 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (-\frac {9}{8} A \,a^{4} b -\frac {5}{4} A \,a^{2} b^{3}-\frac {1}{8} A \,b^{5}+\frac {5}{8} B \,a^{5}+\frac {1}{4} B \,a^{3} b^{2}-\frac {3}{8} B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {b \left (7 A \,a^{4} b +6 A \,a^{2} b^{3}-A \,b^{5}-3 B \,a^{5}+2 B \,a^{3} b^{2}+5 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 a}\right )}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (15 A \,a^{4} b -18 A \,a^{2} b^{3}-A \,b^{5}-3 B \,a^{5}+26 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
| default | \(\frac {\frac {\frac {2 \left (\left (-\frac {9}{8} A \,a^{4} b -\frac {5}{4} A \,a^{2} b^{3}-\frac {1}{8} A \,b^{5}+\frac {5}{8} B \,a^{5}+\frac {1}{4} B \,a^{3} b^{2}-\frac {3}{8} B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {b \left (7 A \,a^{4} b +6 A \,a^{2} b^{3}-A \,b^{5}-3 B \,a^{5}+2 B \,a^{3} b^{2}+5 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 a}\right )}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (15 A \,a^{4} b -18 A \,a^{2} b^{3}-A \,b^{5}-3 B \,a^{5}+26 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
1/d*(2/(a^2+b^2)^3*(((-9/8*A*a^4*b-5/4*A*a^2*b^3-1/8*A*b^5+5/8*B*a^5+1/4*B *a^3*b^2-3/8*B*a*b^4)*cot(d*x+c)^(3/2)-1/8*b*(7*A*a^4*b+6*A*a^2*b^3-A*b^5- 3*B*a^5+2*B*a^3*b^2+5*B*a*b^4)/a*cot(d*x+c)^(1/2))/(b+a*cot(d*x+c))^2+1/8* (15*A*a^4*b-18*A*a^2*b^3-A*b^5-3*B*a^5+26*B*a^3*b^2-3*B*a*b^4)/a/(a*b)^(1/ 2)*arctan(a*cot(d*x+c)^(1/2)/(a*b)^(1/2)))-2/(a^2+b^2)^3*(1/8*(A*a^3-3*A*a *b^2+3*B*a^2*b-B*b^3)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/ (1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/ 2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/8*(3*A*a^2*b-A*b^3-B*a^3+3*B* a*b^2)*2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+2 ^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1 +2^(1/2)*cot(d*x+c)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 8942 vs. \(2 (482) = 964\).
Time = 85.90 (sec) , antiderivative size = 17912, normalized size of antiderivative = 33.54 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {A + B \tan {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \sqrt {\cot {\left (c + d x \right )}}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.02 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, B a^{5} - 15 \, A a^{4} b - 26 \, B a^{3} b^{2} + 18 \, A a^{2} b^{3} + 3 \, B a b^{4} + A b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a 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 {3 \, B a^{3} b - 7 \, A a^{2} b^{2} - 5 \, B a b^{3} + A b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {5 \, B a^{4} - 9 \, A a^{3} b - 3 \, B a^{2} b^{2} - A a b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6} + \frac {2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]
-1/4*((3*B*a^5 - 15*A*a^4*b - 26*B*a^3*b^2 + 18*A*a^2*b^3 + 3*B*a*b^4 + A* b^5)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^7 + 3*a^5*b^2 + 3*a^3*b^ 4 + a*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)/sq rt(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) - ((3*B*a^3*b - 7* A*a^2*b^2 - 5*B*a*b^3 + A*b^4)/sqrt(tan(d*x + c)) + (5*B*a^4 - 9*A*a^3*b - 3*B*a^2*b^2 - A*a*b^3)/tan(d*x + c)^(3/2))/(a^5*b^2 + 2*a^3*b^4 + a*b^6 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)/tan(d*x + c) + (a^7 + 2*a^5*b^2 + a^3*b^4 )/tan(d*x + c)^2))/d
\[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]