Optimal. Leaf size=320 \[ \frac {2 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}+\frac {a \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}-\frac {a \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A]
time = 0.19, antiderivative size = 320, normalized size of antiderivative = 1.00, number of
steps used = 17, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used
= {3985, 3966, 3969, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2694, 2653, 2720}
\begin {gather*} \frac {a \text {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 \text {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 \cot (c+d x) (a \sec (c+d x)+3 a)}{3 d (e \cot (c+d x))^{3/2}}+\frac {a \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 \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 \sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{3 d (e \cot (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2653
Rule 2694
Rule 2720
Rule 3557
Rule 3966
Rule 3969
Rule 3985
Rubi steps
\begin {align*} \int \frac {a+a \sec (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx &=\frac {\int (a+a \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {2 \int \frac {\frac {3 a}{2}+\frac {1}{2} a \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 {2 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \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 \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{(e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {\left (a \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 \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {\left (a \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 {(2 a) \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \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 \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \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 \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 \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 \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a \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 \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 \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 \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 \cot (c+d x) (3 a+a \sec (c+d x))}{3 d (e \cot (c+d x))^{3/2}}-\frac {a \cot (c+d x) \csc (c+d x) F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {\sin (2 c+2 d x)}}{3 d (e \cot (c+d x))^{3/2}}+\frac {a \tan ^{-1}\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 \tan ^{-1}\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 \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 \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)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 12.66, size = 224, normalized size = 0.70 \begin {gather*} \frac {a (1+\cos (c+d x)) \cos (2 (c+d x)) \csc (c+d x) \sqrt {\csc ^2(c+d x)} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (-4 \sqrt [4]{-1} \cot ^{\frac {3}{2}}(c+d x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\cot (c+d x)}\right )\right |-1\right )+\sqrt {\csc ^2(c+d x)} \left (4+12 \cos (c+d x)+3 \text {ArcSin}(\cos (c+d x)-\sin (c+d x)) \cot (c+d x) \sqrt {\sin (2 (c+d x))}-3 \cot (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 (e \cot (c+d x))^{3/2} \left (-1+\cot ^2(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.26, size = 698, normalized size = 2.18
method | result | size |
default | \(-\frac {a \left (-1+\cos \left (d x +c \right )\right ) \left (-3 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+4 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-6 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+4 \sqrt {2}\, \cos \left (d x +c \right )+2 \sqrt {2}\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \sqrt {2}}{6 d \left (\frac {e \cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{5}}\) | \(698\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.24, size = 125, normalized size = 0.39 \begin {gather*} \frac {{\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 8 \, \sqrt {\tan \left (d x + c\right )}\right )} a e^{\left (-\frac {3}{2}\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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