Optimal. Leaf size=106 \[ \frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}+\frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {14 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 106, 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 = {2670, 2680, 2682, 2640, 2639} \[ \frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac {14 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2670
Rule 2680
Rule 2682
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {a^6 \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^3} \, dx}{e^6}\\ &=\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac {\left (7 a^4\right ) \int \frac {(e \cos (c+d x))^{5/2}}{a-a \sin (c+d x)} \, dx}{e^4}\\ &=\frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac {\left (7 a^3\right ) \int \sqrt {e \cos (c+d x)} \, dx}{e^2}\\ &=\frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac {\left (7 a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{e^2 \sqrt {\cos (c+d x)}}\\ &=\frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {14 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 64, normalized size = 0.60 \[ \frac {8\ 2^{3/4} a^3 \sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac {7}{4},-\frac {1}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 146, normalized size = 1.38 \[ -\frac {2 \left (-4 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-24 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-13 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{3 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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