Optimal. Leaf size=154 \[ -\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.18, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2680, 2681, 2683, 2642, 2641} \[ -\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2680
Rule 2681
Rule 2683
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx}{11 a^2}\\ &=-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {\left (3 e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))} \, dx}{77 a^3}\\ &=-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{77 a^4}\\ &=-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 a^4 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 66, normalized size = 0.43 \[ -\frac {(e \cos (c+d x))^{5/2} \, _2F_1\left (\frac {5}{4},\frac {15}{4};\frac {9}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{10\ 2^{3/4} a^4 d e (\sin (c+d x)+1)^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} e \cos \left (d x + c\right )}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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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maple [B] time = 4.37, size = 583, normalized size = 3.79 \[ \frac {2 \left (32 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+80 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-176 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-78 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+176 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{2}}{77 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, 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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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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