Optimal. Leaf size=157 \[ \frac {2 a \left (a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {6 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e} \]
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Rubi [A]
time = 0.16, antiderivative size = 157, 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 = {2771, 2941,
2748, 2721, 2720} \begin {gather*} \frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 a \left (a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}{5 d e}+\frac {6 a b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}{5 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rule 2941
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3}{\sqrt {e \sin (c+d x)}} \, dx &=\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2}{5} \int \frac {(a+b \cos (c+d x)) \left (\frac {5 a^2}{2}+2 b^2+\frac {9}{2} a b \cos (c+d x)\right )}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {6 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e}+\frac {4}{15} \int \frac {\frac {15}{4} a \left (a^2+2 b^2\right )+\frac {3}{4} b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {6 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e}+\left (a \left (a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {6 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e}+\frac {\left (a \left (a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{\sqrt {e \sin (c+d x)}}\\ &=\frac {2 a \left (a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {6 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{5 d e}+\frac {2 b (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}{5 d e}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 98, normalized size = 0.62 \begin {gather*} \frac {-10 a \left (a^2+2 b^2\right ) F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sqrt {\sin (c+d x)}+b \left (30 a^2+9 b^2+10 a b \cos (c+d x)+b^2 \cos (2 (c+d x))\right ) \sin (c+d x)}{5 d \sqrt {e \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 210, normalized size = 1.34
method | result | size |
default | \(-\frac {5 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{3}+10 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a \,b^{2}-2 b^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 a \,b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-30 a^{2} b \cos \left (d x +c \right ) \sin \left (d x +c \right )-8 b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(210\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 126, normalized size = 0.80 \begin {gather*} \frac {{\left (5 \, \sqrt {2} \sqrt {-i} {\left (a^{3} + 2 \, a b^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i} {\left (a^{3} + 2 \, a b^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} + 5 \, a b^{2} \cos \left (d x + c\right ) + 15 \, a^{2} b + 4 \, b^{3}\right )} \sqrt {\sin \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx \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 \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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