Optimal. Leaf size=165 \[ \frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^6 (e \cos (c+d x))^{3/2}}{15 d e^7 \left (a^3-a^3 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2670, 2681, 2683, 2640, 2639} \[ \frac {2 a^6 (e \cos (c+d x))^{3/2}}{15 d e^7 \left (a^3-a^3 \sin (c+d x)\right )}+\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d e^6 \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2670
Rule 2681
Rule 2683
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{11/2}} \, dx &=\frac {a^6 \int \frac {\sqrt {e \cos (c+d x)}}{(a-a \sin (c+d x))^3} \, dx}{e^6}\\ &=\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {a^5 \int \frac {\sqrt {e \cos (c+d x)}}{(a-a \sin (c+d x))^2} \, dx}{3 e^6}\\ &=\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}+\frac {a^4 \int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)} \, dx}{15 e^6}\\ &=\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}+\frac {2 a^4 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))}-\frac {a^3 \int \sqrt {e \cos (c+d x)} \, dx}{15 e^6}\\ &=\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}+\frac {2 a^4 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))}-\frac {\left (a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 e^6 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac {2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}+\frac {2 a^4 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 66, normalized size = 0.40 \[ \frac {2\ 2^{3/4} a^3 (\sin (c+d x)+1)^{9/4} \, _2F_1\left (-\frac {9}{4},\frac {1}{4};-\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{6} \cos \left (d x + c\right )^{6}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.68, size = 514, normalized size = 3.12 \[ -\frac {2 \left (48 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \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}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-96 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \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}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+72 \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}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-152 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \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}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+36 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \EllipticE \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}-48 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-36 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{45 \left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \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}\, e^{5} 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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}} \,d x \]
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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