Optimal. Leaf size=152 \[ \frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac {10 a^4 \sin (c+d x) \sqrt {e \cos (c+d x)}}{d e^3}-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}}+\frac {12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^4-a^4 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 152, 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, 2642, 2641} \[ \frac {12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^4-a^4 \sin (c+d x)\right )}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac {10 a^4 \sin (c+d x) \sqrt {e \cos (c+d x)}}{d e^3}-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2670
Rule 2680
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {a^8 \int \frac {(e \cos (c+d x))^{11/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8}\\ &=\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac {\left (3 a^6\right ) \int \frac {(e \cos (c+d x))^{7/2}}{(a-a \sin (c+d x))^2} \, dx}{e^6}\\ &=\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (15 a^4\right ) \int (e \cos (c+d x))^{3/2} \, dx}{e^4}\\ &=-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (5 a^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{e^2}\\ &=-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (5 a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{e^2 \sqrt {e \cos (c+d x)}}\\ &=-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}}-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 66, normalized size = 0.43 \[ \frac {16 \sqrt [4]{2} a^4 (\sin (c+d x)+1)^{3/4} \, _2F_1\left (-\frac {9}{4},-\frac {3}{4};\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.28, size = 263, normalized size = 1.73 \[ \frac {2 \left (-8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-48 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-18 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d} \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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