Optimal. Leaf size=189 \[ \frac {4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac {4 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {4 e \sqrt {e \sin (c+d x)}}{a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d} \]
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Rubi [A]
time = 0.42, 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 = {3957, 2954,
2952, 2647, 2721, 2720, 2644, 14, 2649} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2644
Rule 2647
Rule 2649
Rule 2720
Rule 2721
Rule 2952
Rule 2954
Rule 3957
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 ) \text {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 ) \text {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 = 1.24, size = 119, normalized size = 0.63 \begin {gather*} \frac {2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left ((15+10 \cos (c+d x)-\cos (2 (c+d x))) \csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {24 F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )}{\sin ^{\frac {3}{2}}(c+d x)}\right ) (e \sin (c+d x))^{3/2}}{3 a^2 d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 153, normalized size = 0.81
method | result | size |
default | \(-\frac {2 e^{3} \left (3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\left (\cos ^{6}\left (d x +c \right )\right )+6 \left (\cos ^{5}\left (d x +c \right )\right )+4 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{3}\left (d x +c \right )\right )-3 \left (\cos ^{2}\left (d x +c \right )\right )+8 \cos \left (d x +c \right )\right )}{3 a^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )-1\right ) d}\) | \(153\) |
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 = 0.79, size = 141, normalized size = 0.75 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {-i} {\left (\sqrt {2} \cos \left (d x + c\right ) e^{\frac {3}{2}} + \sqrt {2} e^{\frac {3}{2}}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {i} {\left (\sqrt {2} \cos \left (d x + c\right ) e^{\frac {3}{2}} + \sqrt {2} e^{\frac {3}{2}}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (\cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - 5 \, \cos \left (d x + c\right ) e^{\frac {3}{2}} - 8 \, e^{\frac {3}{2}}\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{3 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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 \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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