Optimal. Leaf size=160 \[ -\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (c-d (5+4 n)) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2842, 21, 2855,
72, 71} \begin {gather*} \frac {2 a^2 (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 71
Rule 72
Rule 2842
Rule 2855
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^n \, dx &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {(c+d \sin (e+f x))^n \left (-\frac {1}{2} a^2 (c-5 d-4 d n)-\frac {1}{2} a^2 (c-5 d-4 d n) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{d (3+2 n)}\\ &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-d (5+4 n))) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{d (3+2 n)}\\ &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {\left (a^3 (c-d (5+4 n)) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {\left (a^3 (c-d (5+4 n)) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (\frac {c}{c+d}+\frac {d x}{c+d}\right )^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (c-d (5+4 n)) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 4.29, size = 133, normalized size = 0.83 \begin {gather*} -\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \left ((-c+d (5+4 n)) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {d (-1+\sin (e+f x))}{c+d}\right )+(c+d) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{1+n}\right )}{d f (3+2 n) \sqrt {a (1+\sin (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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