Integrand size = 23, antiderivative size = 322 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^4-26 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {a} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{4 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))} \] Output:
1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2 +b^2)^3/d+1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^( 1/2)/(a^2+b^2)^3/d-1/4*(3*a^4-26*a^2*b^2+3*b^4)*arctan(a^(1/2)*cot(d*x+c)^ (1/2)/b^(1/2))/a^(1/2)/b^(1/2)/(a^2+b^2)^3/d-1/2*(a-b)*(a^2+4*a*b+b^2)*arc tanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/(a^2+b^2)^3/d-1/2*b* cot(d*x+c)^(1/2)/(a^2+b^2)/d/(b+a*cot(d*x+c))^2+1/4*(5*a^2-3*b^2)*cot(d*x+ c)^(1/2)/(a^2+b^2)^2/d/(b+a*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.10 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {-\frac {2 b^{3/2} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\sqrt {a} \left (a^2+b^2\right )^3}+\frac {2 b \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{\left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 \left (a^2+b^2\right )^3}-\frac {-\frac {3 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right ) \sqrt {\cot (c+d x)}}{\sqrt {b}}+\frac {2 a^2 \cot ^2(c+d x)}{(b+a \cot (c+d x))^2}+\frac {3 a \cot (c+d x)}{b+a \cot (c+d x)}}{4 a \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}+\frac {2 a \left (a^2-3 b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-\cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {4 a^2 \cot ^{\frac {5}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\frac {a \cot (c+d x)}{b}\right )}{5 b \left (a^2+b^2\right )^2}-\frac {b \left (3 a^2-b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 \left (a^2+b^2\right )^3}}{d} \] Input:
Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]
Output:
-(((-2*b^(3/2)*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]) /(Sqrt[a]*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/(a^2 + b ^2)^3 - (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*(a^2 + b^2)^3) - ((-3*Sq rt[a]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]*Sqrt[Cot[c + d*x]])/Sqr t[b] + (2*a^2*Cot[c + d*x]^2)/(b + a*Cot[c + d*x])^2 + (3*a*Cot[c + d*x])/ (b + a*Cot[c + d*x]))/(4*a*(a^2 + b^2)*Sqrt[Cot[c + d*x]]) + (2*a*(a^2 - 3 *b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7 /4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(5/2)*Hyper geometric2F1[2, 5/2, 7/2, -((a*Cot[c + d*x])/b)])/(5*b*(a^2 + b^2)^2) - (b *(3*a^2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[ 2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2] *Log[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]]))/(4*(a^2 + b^2)^3))/d)
Time = 1.81 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.11, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.130, Rules used = {3042, 4156, 3042, 4050, 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.
| \(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (c+d x)^{3/2} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4050 |
| \(\displaystyle -\frac {\int \frac {-3 b \cot ^2(c+d x)-4 a \cot (c+d x)+b}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-3 b \cot ^2(c+d x)-4 a \cot (c+d x)+b}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-3 b \tan \left (c+d x+\frac {\pi }{2}\right )^2+4 a \tan \left (c+d x+\frac {\pi }{2}\right )+b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {-\frac {\int \frac {16 a \cot (c+d x) b^2-\left (5 a^2-3 b^2\right ) \cot ^2(c+d x) b+\left (3 a^2-5 b^2\right ) b}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {16 a \cot (c+d x) b^2-\left (5 a^2-3 b^2\right ) \cot ^2(c+d x) b+\left (3 a^2-5 b^2\right ) b}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {-16 a \tan \left (c+d x+\frac {\pi }{2}\right ) b^2-\left (5 a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 b+\left (3 a^2-5 b^2\right ) b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {8 \left (b^2 \left (3 a^2-b^2\right )-a b \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {8 \int \frac {b^2 \left (3 a^2-b^2\right )-a b \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {8 \int \frac {\left (3 a^2-b^2\right ) b^2+a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {-\frac {\frac {16 \int -\frac {b \left (b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int \frac {b \left (b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {-\frac {\frac {b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {-\frac {-\frac {2 b \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 \left (a^2+b^2\right )}-\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {-\frac {\left (5 a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\frac {2 \sqrt {b} \left (3 a^4-26 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {16 b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}}{4 \left (a^2+b^2\right )}\) |
Input:
Int[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]
Output:
-1/2*(b*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (-((( 5*a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x]))) - ((2*Sqrt[b]*(3*a^4 - 26*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sq rt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - (16*b*(-1/2*((a + b)*(a^2 - 4*a*b + b^2) *(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*S qrt[Cot[c + d*x]]]/Sqrt[2])) + ((a - b)*(a^2 + 4*a*b + b^2)*(-1/2*Log[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*b*(a^2 + b^2)))/(4*(a^2 + b^2))
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ (n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 *a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^ 2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 2*m]
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]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(2040\) vs. \(2(284)=568\).
Time = 0.28 (sec) , antiderivative size = 2041, normalized size of antiderivative = 6.34
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2041\) |
| default | \(\text {Expression too large to display}\) | \(2041\) |
Input:
int(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/4/d*(-2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/2)*2^(1/2)*a^5+6*ar ctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a*b^5*tan(d*x+c)+3*arctan(b*tan(d*x+c )^(1/2)/(a*b)^(1/2))*a^4*b^2*tan(d*x+c)^2-26*arctan(b*tan(d*x+c)^(1/2)/(a* b)^(1/2))*a^2*b^4*tan(d*x+c)^2-ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1) /(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*(a*b)^(1/2)*2^(1/2)*a^5-2*arctan (1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/2)*2^(1/2)*a^5-3*tan(d*x+c)^(1/2)*(a *b)^(1/2)*a*b^4-2*tan(d*x+c)^(3/2)*(a*b)^(1/2)*a^2*b^3+3*tan(d*x+c)^(3/2)* (a*b)^(1/2)*a^4*b+6*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^5*b*tan(d*x+c )-52*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^3*b^3*tan(d*x+c)+2*tan(d*x+c )^(1/2)*(a*b)^(1/2)*a^3*b^2+6*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/ 2)*2^(1/2)*a*b^4*tan(d*x+c)^2-2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^ (1/2)*2^(1/2)*a^3*b^2*tan(d*x+c)^2+6*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*( a*b)^(1/2)*2^(1/2)*a^2*b^3*tan(d*x+c)^2+6*arctan(-1+2^(1/2)*tan(d*x+c)^(1/ 2))*(a*b)^(1/2)*2^(1/2)*a*b^4*tan(d*x+c)^2+3*ln(-(2^(1/2)*tan(d*x+c)^(1/2) -tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*(a*b)^(1/2)*2^(1/2 )*a^2*b^3*tan(d*x+c)^2-2*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1 /2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*(a*b)^(1/2)*2^(1/2)*a^4*b*tan(d*x+c)-4 *arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/2)*2^(1/2)*a^4*b*tan(d*x+c)+1 2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/2)*2^(1/2)*a^3*b^2*tan(d*x+c )-4*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(a*b)^(1/2)*2^(1/2)*a^4*b*tan(d...
Leaf count of result is larger than twice the leaf count of optimal. 2409 vs. \(2 (284) = 568\).
Time = 0.48 (sec) , antiderivative size = 4843, normalized size of antiderivative = 15.04 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/cot(d*x+c)**(3/2)/(a+b*tan(d*x+c))**3,x)
Output:
Integral(1/((a + b*tan(c + d*x))**3*cot(c + d*x)**(3/2)), x)
Time = 0.25 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, a^{4} - 26 \, a^{2} b^{2} + 3 \, b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {3 \, a^{2} b - 5 \, b^{3}}{\sqrt {\tan \left (d x + c\right )}} + \frac {5 \, a^{3} - 3 \, a b^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + \frac {2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \] Input:
integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
-1/4*((3*a^4 - 26*a^2*b^2 + 3*b^4)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))) )/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a*b)) - (2*sqrt(2)*(a^3 - 3*a^ 2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt (2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*(a^3 + 3*a^2*b - 3* a*b^2 - 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*a^2*b - 5*b^3)/sqrt(tan(d*x + c)) + (5 *a^3 - 3*a*b^2)/tan(d*x + c)^(3/2))/(a^4*b^2 + 2*a^2*b^4 + b^6 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)/tan(d*x + c) + (a^6 + 2*a^4*b^2 + a^2*b^4)/tan(d*x + c)^2))/d
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
integrate(1/((b*tan(d*x + c) + a)^3*cot(d*x + c)^(3/2)), x)
Timed out. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \] Input:
int(1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3),x)
Output:
int(1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3), x)
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3} b^{3}+3 \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2} a \,b^{2}+3 \cot \left (d x +c \right )^{2} \tan \left (d x +c \right ) a^{2} b +\cot \left (d x +c \right )^{2} a^{3}}d x \] Input:
int(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x)
Output:
int(sqrt(cot(c + d*x))/(cot(c + d*x)**2*tan(c + d*x)**3*b**3 + 3*cot(c + d *x)**2*tan(c + d*x)**2*a*b**2 + 3*cot(c + d*x)**2*tan(c + d*x)*a**2*b + co t(c + d*x)**2*a**3),x)