Optimal. Leaf size=189 \[ \frac {4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {4 e \sqrt {e \sin (c+d x)}}{a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d} \]
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Rubi [A] time = 0.59, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3872, 2875, 2873, 2567, 2642, 2641, 2564, 14, 2569} \[ \frac {4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {4 e \sqrt {e \sin (c+d x)}}{a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2569
Rule 2641
Rule 2642
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{3/2}}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx}{a^4}\\ &=\frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{5/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{5/2}}\right ) \, dx}{a^4}\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{a^2}\\ &=-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {\left (2 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^2}-\frac {\left (2 e^2\right ) \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{5/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d}-\frac {\left (4 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}-\frac {1}{e^2 \sqrt {x}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}-\frac {\left (2 e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 \sqrt {e \sin (c+d x)}}\\ &=\frac {4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 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 a^2 d \sqrt {e \sin (c+d x)}}+\frac {4 e \sqrt {e \sin (c+d x)}}{a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d}-\frac {\left (4 e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 \sqrt {e \sin (c+d x)}}\\ &=\frac {4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {4 e \sqrt {e \sin (c+d x)}}{a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.91, size = 119, normalized size = 0.63 \[ \frac {2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) (e \sin (c+d x))^{3/2} \left (\frac {24 F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac {3}{2}}(c+d x)}+(10 \cos (c+d x)-\cos (2 (c+d x))+15) \csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )}{3 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \sin \left (d x + c\right )} e \sin \left (d x + c\right )}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.30, size = 153, normalized size = 0.81 \[ -\frac {2 e^{3} \left (3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\left (\cos ^{6}\left (d x +c \right )\right )+6 \left (\cos ^{5}\left (d x +c \right )\right )+4 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{3}\left (d x +c \right )\right )-3 \left (\cos ^{2}\left (d x +c \right )\right )+8 \cos \left (d x +c \right )\right )}{3 a^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )-1\right ) d} \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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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