Optimal. Leaf size=114 \[ \frac {2 \left (3 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e} \]
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Rubi [A]
time = 0.09, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2771, 2748,
2721, 2720} \begin {gather*} \frac {2 \left (3 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 )}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2}{\sqrt {e \sin (c+d x)}} \, dx &=\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2}{3} \int \frac {\frac {3 a^2}{2}+b^2+\frac {5}{2} a b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {1}{3} \left (3 a^2+2 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {\left (\left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 \sqrt {e \sin (c+d x)}}\\ &=\frac {2 \left (3 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)}}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 79, normalized size = 0.69 \begin {gather*} \frac {-2 \left (3 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)}+2 b (6 a+b \cos (c+d x)) \sin (c+d x)}{3 d \sqrt {e \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 170, normalized size = 1.49
method | result | size |
default | \(-\frac {3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-12 a b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(170\) |
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 = 0.10, size = 105, normalized size = 0.92 \begin {gather*} \frac {{\left (\sqrt {2} \sqrt {-i} {\left (3 \, a^{2} + 2 \, b^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} \sqrt {i} {\left (3 \, a^{2} + 2 \, 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^{2} \cos \left (d x + c\right ) + 6 \, a b\right )} \sqrt {\sin \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, 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 )^{2}}{\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 )}^2}{\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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