Integrand size = 25, antiderivative size = 375 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}} \]
[Out]
Time = 0.43 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3985, 3971, 3554, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2691, 2694, 2653, 2720, 2687, 30} \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 d (e \cot (c+d x))^{3/2}} \]
[In]
[Out]
Rule 30
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2653
Rule 2687
Rule 2691
Rule 2694
Rule 2720
Rule 3554
Rule 3557
Rule 3971
Rule 3985
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sec (c+d x))^2 \tan ^{\frac {3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {\int \left (a^2 \tan ^{\frac {3}{2}}(c+d x)+2 a^2 \sec (c+d x) \tan ^{\frac {3}{2}}(c+d x)+a^2 \sec ^2(c+d x) \tan ^{\frac {3}{2}}(c+d x)\right ) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {a^2 \int \tan ^{\frac {3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \int \sec ^2(c+d x) \tan ^{\frac {3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\left (2 a^2\right ) \int \sec (c+d x) \tan ^{\frac {3}{2}}(c+d x) \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {\left (2 a^2\right ) \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \text {Subst}\left (\int x^{3/2} \, dx,x,\tan (c+d x)\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac {\left (2 a^2 \cos ^{\frac {3}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 (e \cot (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac {\left (2 a^2 \cot (c+d x) \csc (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{3 (e \cot (c+d x))^{3/2}}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac {a^2 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}-\frac {a^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.34 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\frac {a^2 \left (\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )+2 \left (5 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},-\tan ^2(c+d x)\right )\right )\right ) (1+\sec (c+d x))^2 \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right ) \sin ^2(c+d x)}{10 d e \sqrt {e \cot (c+d x)}} \]
[In]
[Out]
Time = 11.11 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.08
method | result | size |
parts | \(-\frac {2 a^{2} e \left (-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}+\frac {2 a^{2} e}{5 d \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2 a^{2} \sqrt {2}\, \left (-\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{2}-\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+\sqrt {2}\, \sin \left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d e \sqrt {e \cot \left (d x +c \right )}\, \left (\cos \left (d x +c \right )^{2}-1\right )}\) | \(405\) |
default | \(\text {Expression too large to display}\) | \(933\) |
[In]
[Out]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=a^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
[In]
[Out]
Exception generated. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]