Optimal. Leaf size=192 \[ \frac {2 a^2 e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {a^2 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d} \]
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Rubi [A]
time = 0.26, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3957, 2952,
2715, 2721, 2720, 2644, 327, 335, 218, 212, 209, 2646} \begin {gather*} \frac {2 a^2 e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {a^2 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 327
Rule 335
Rule 2644
Rule 2646
Rule 2715
Rule 2720
Rule 2721
Rule 2952
Rule 3957
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx &=\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{3/2} \, dx\\ &=\int \left (a^2 (e \sin (c+d x))^{3/2}+2 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}+a^2 \sec ^2(c+d x) (e \sin (c+d x))^{3/2}\right ) \, dx\\ &=a^2 \int (e \sin (c+d x))^{3/2} \, dx+a^2 \int \sec ^2(c+d x) (e \sin (c+d x))^{3/2} \, dx+\left (2 a^2\right ) \int \sec (c+d x) (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {1}{3} \left (a^2 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx-\frac {1}{2} \left (a^2 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d}+\frac {\left (2 a^2 e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d}+\frac {\left (a^2 e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 \sqrt {e \sin (c+d x)}}-\frac {\left (a^2 e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {a^2 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d}+\frac {\left (4 a^2 e\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=-\frac {a^2 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d}+\frac {\left (2 a^2 e^2\right ) \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {\left (2 a^2 e^2\right ) \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=\frac {2 a^2 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {a^2 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {4 a^2 e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a^2 e \sec (c+d x) \sqrt {e \sin (c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 30.40, size = 220, normalized size = 1.15 \begin {gather*} \frac {16 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt {e \sin (c+d x)} \left (6 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}-\sqrt {\cos ^2(c+d x)} F\left (\left .\text {ArcSin}\left (\sqrt {\sin (c+d x)}\right )\right |-1\right )-3 \sqrt {\cos ^2(c+d x)} \log \left (1-\sqrt {\sin (c+d x)}\right )+3 \sqrt {\cos ^2(c+d x)} \log \left (1+\sqrt {\sin (c+d x)}\right )+\sqrt {\sin (c+d x)}-12 \sqrt {\cos ^2(c+d x)} \sqrt {\sin (c+d x)}+2 \sin ^{\frac {5}{2}}(c+d x)\right ) \sin ^4\left (\frac {1}{2} \text {ArcSin}(\sin (c+d x))\right )}{3 d \sin ^{\frac {9}{2}}(c+d x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.22, size = 201, normalized size = 1.05
method | result | size |
default | \(\frac {a^{2} \left (12 \cos \left (d x +c \right ) e^{\frac {3}{2}} \sqrt {e \sin \left (d x +c \right )}\, \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+12 \cos \left (d x +c \right ) e^{\frac {3}{2}} \sqrt {e \sin \left (d x +c \right )}\, \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+\left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, e^{2}-4 \left (\cos ^{2}\left (d x +c \right )\right ) e^{2} \sin \left (d x +c \right )-24 e^{2} \sin \left (d x +c \right ) \cos \left (d x +c \right )+6 e^{2} \sin \left (d x +c \right )\right )}{6 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(201\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 9.16, size = 279, normalized size = 1.45 \begin {gather*} -\frac {\sqrt {2} \sqrt {-i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} \sqrt {i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 6 \, a^{2} \arctan \left (-\frac {2 \, {\left (76 \, \cos \left (d x + c\right )^{2} - 425 \, {\left (\sin \left (d x + c\right ) - 1\right )} \sqrt {\sin \left (d x + c\right )} - 152 \, \sin \left (d x + c\right ) - 152\right )}}{361 \, \cos \left (d x + c\right )^{2} + 978 \, \sin \left (d x + c\right ) - 722}\right ) \cos \left (d x + c\right ) e^{\frac {3}{2}} - 3 \, a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} \log \left (\frac {\cos \left (d x + c\right )^{2} - 4 \, {\left (\sin \left (d x + c\right ) + 1\right )} \sqrt {\sin \left (d x + c\right )} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) + 2 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 12 \, a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} - 3 \, a^{2} e^{\frac {3}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{6 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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