Integrand size = 25, antiderivative size = 359 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=-\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {2 \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.50 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3985, 3973, 3971, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2689, 2694, 2653, 2720, 2687, 30} \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a^2 d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a^2 d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {2 \sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 a^2 d (e \cot (c+d x))^{3/2}} \]
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Rule 30
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2653
Rule 2687
Rule 2689
Rule 2694
Rule 2720
Rule 3555
Rule 3557
Rule 3971
Rule 3973
Rule 3985
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {\int \frac {(-a+a \sec (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{a^4 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {\int \left (\frac {a^2}{\tan ^{\frac {5}{2}}(c+d x)}-\frac {2 a^2 \sec (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}\right ) \, dx}{a^4 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = \frac {\int \frac {1}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\int \frac {\sec ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {\sec (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {2 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {2 \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\tan (c+d x)\right )}{a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\left (2 \cos ^{\frac {3}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 a^2 (e \cot (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\left (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 a^2 (e \cot (c+d x))^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {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 a^2 d (e \cot (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {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 a^2 d (e \cot (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\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} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\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} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {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 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac {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 a^2 d (e \cot (c+d x))^{3/2}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx \]
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Result contains complex when optimal does not.
Time = 9.01 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.78
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{2} \left (3 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-10 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+4 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-4 \csc \left (d x +c \right )+4 \cot \left (d x +c \right )\right ) \sin \left (d x +c \right )}{6 a^{2} d {\left (-\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {3}{2}} \left (1-\cos \left (d x +c \right )\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )}}\) | \(638\) |
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Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{2}{\left (c + d x \right )} + 2 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \]
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Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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