Optimal. Leaf size=156 \[ \frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac {154 a^4 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac {154 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^2 \sqrt {\cos (c+d x)}}+\frac {44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^4-a^4 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.23, antiderivative size = 156, 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, 2680, 2635, 2640, 2639} \[ \frac {44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^4-a^4 \sin (c+d x)\right )}+\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac {154 a^4 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac {154 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^2 \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2670
Rule 2680
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {a^8 \int \frac {(e \cos (c+d x))^{13/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8}\\ &=\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac {\left (11 a^6\right ) \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^2} \, dx}{e^6}\\ &=\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}+\frac {44 a^6 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (77 a^4\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 e^4}\\ &=-\frac {154 a^4 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d e^3}+\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}+\frac {44 a^6 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (77 a^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^2}\\ &=-\frac {154 a^4 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d e^3}+\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}+\frac {44 a^6 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (77 a^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^2 \sqrt {\cos (c+d x)}}\\ &=-\frac {154 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)}}-\frac {154 a^4 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d e^3}+\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}+\frac {44 a^6 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 64, normalized size = 0.41 \[ \frac {16\ 2^{3/4} a^4 \sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac {11}{4},-\frac {1}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.15, size = 190, normalized size = 1.22 \[ -\frac {2 \left (-24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-246 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+80 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{15 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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