Integrand size = 23, antiderivative size = 346 \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=-\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}-\frac {2 a (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a (e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {a (e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d} \]
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Time = 0.50 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3985, 3967, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719} \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\frac {a \tan ^{\frac {3}{2}}(c+d x) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{\frac {3}{2}}(c+d x) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac {2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac {2 a \sin (c+d x) \tan (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) (e \cot (c+d x))^{3/2}}{d \sqrt {\sin (2 c+2 d x)}} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2652
Rule 2693
Rule 2695
Rule 2719
Rule 3557
Rule 3967
Rule 3969
Rule 3985
Rubi steps \begin{align*} \text {integral}& = \left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {a+a \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}+\left (2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \left (-\frac {a}{2}+\frac {1}{2} a \sec (c+d x)\right ) \sqrt {\tan (c+d x)} \, dx \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}-\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\tan (c+d x)} \, dx+\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \sec (c+d x) \sqrt {\tan (c+d x)} \, dx \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d}-\left (2 a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {\left (2 a (e \cot (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{\cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (2 a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {\left (2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan (c+d x)\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{\sqrt {\sin (2 c+2 d x)}}+\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}-\frac {2 a (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \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}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \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} \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}-\frac {2 a (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{d \sqrt {\sin (2 c+2 d x)}}-\frac {a (e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {a (e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {2 (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \tan (c+d x)}{d}-\frac {2 a (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a (e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {a (e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {2 a (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.24 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.55 \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\frac {a e (1+\cos (c+d x)) \sqrt {e \cot (c+d x)} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (8 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\cot ^2(c+d x)\right )+3 \sqrt {\csc ^2(c+d x)} \left (-4 \cos (c+d x)-4 \cos ^2(c+d x)+\arcsin (\cos (c+d x)-\sin (c+d x)) \sqrt {\sin (2 (c+d x))}+\log \left (\cos (c+d x)+\sin (c+d x)+\sqrt {\sin (2 (c+d x))}\right ) \sqrt {\sin (2 (c+d x))}\right )\right )}{12 d \sqrt {\csc ^2(c+d x)}} \]
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Time = 10.76 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.45
method | result | size |
parts | \(-\frac {2 a e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \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}\right )}{d}-\frac {a \sqrt {2}\, e \sqrt {e \cot \left (d x +c \right )}\, \left (-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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\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 )-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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (d x +c \right )+\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 ) \sec \left (d x +c \right )+\sqrt {2}\right )}{d}\) | \(501\) |
default | \(\text {Expression too large to display}\) | \(1035\) |
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Timed out. \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=a \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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