Optimal. Leaf size=154 \[ \frac {a e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d} \]
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Rubi [A] time = 0.20, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3872, 2838, 2564, 321, 329, 212, 206, 203, 2635, 2642, 2641} \[ \frac {a e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 321
Rule 329
Rule 2564
Rule 2635
Rule 2641
Rule 2642
Rule 2838
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx &=-\int (-a-a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^{3/2} \, dx\\ &=a \int (e \sin (c+d x))^{3/2} \, dx+a \int \sec (c+d x) (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a \operatorname {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 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {(a e) \operatorname {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 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 {2 a 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 {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {(2 a e) \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=\frac {2 a 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 {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {\left (a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {\left (a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=\frac {a e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a 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 {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 170, normalized size = 1.10 \[ \frac {a (e \sin (c+d x))^{3/2} \left (-24 \sqrt {\sin (c+d x)}-8 F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )-3 \log \left (1-\sqrt {\sin (c+d x)}\right )+3 \log \left (\sqrt {\sin (c+d x)}+1\right )+12 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )+6 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )+16 \sin ^{\frac {5}{2}}(c+d x) \cos (c+d x) \sec (2 (c+d x))-8 \sqrt {\sin (c+d x)} \cos (c+d x) \sec (2 (c+d x))\right )}{12 d \sin ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e \sec \left (d x + c\right ) + a e\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.63, size = 210, normalized size = 1.36 \[ \frac {a \,e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{d}+\frac {a \,e^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{d}-\frac {2 a e \sqrt {e \sin \left (d x +c \right )}}{d}-\frac {a \,e^{2} \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{3 d \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}+\frac {2 a \,e^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}-\frac {2 a \,e^{2} \sin \left (d x +c \right )}{3 d \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}} \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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