Optimal. Leaf size=235 \[ -\frac {2 (3 b c+5 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {2 \left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (3 b c+5 a d) \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.24, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2832, 2831,
2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 \left (c^2-d^2\right ) (5 a d+3 b c) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (5 a d+3 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rubi steps
\begin {align*} \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {2}{5} \int \sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} (5 a c+3 b d)+\frac {1}{2} (3 b c+5 a d) \sin (e+f x)\right ) \, dx\\ &=-\frac {2 (3 b c+5 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {4}{15} \int \frac {\frac {1}{4} \left (12 b c d+5 a \left (3 c^2+d^2\right )\right )+\frac {1}{4} \left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx\\ &=-\frac {2 (3 b c+5 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {\left ((3 b c+5 a d) \left (c^2-d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d}+\frac {\left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{15 d}\\ &=-\frac {2 (3 b c+5 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {\left (\left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{15 d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left ((3 b c+5 a d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{15 d \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 (3 b c+5 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {2 \left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (3 b c+5 a d) \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 218, normalized size = 0.93 \begin {gather*} \frac {-2 d \left (12 b c d+5 a \left (3 c^2+d^2\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-2 \left (20 a c d+3 b \left (c^2+3 d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-2 d \cos (e+f x) (c+d \sin (e+f x)) (6 b c+5 a d+3 b d \sin (e+f x))}{15 d f \sqrt {c+d \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1448\) vs.
\(2(281)=562\).
time = 7.01, size = 1449, normalized size = 6.17
method | result | size |
default | \(\text {Expression too large to display}\) | \(1449\) |
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 = 0.16, size = 521, normalized size = 2.22 \begin {gather*} -\frac {\sqrt {2} {\left (6 \, b c^{3} - 5 \, a c^{2} d - 18 \, b c d^{2} - 15 \, a d^{3}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (6 \, b c^{3} - 5 \, a c^{2} d - 18 \, b c d^{2} - 15 \, a d^{3}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (3 i \, b c^{2} d + 20 i \, a c d^{2} + 9 i \, b d^{3}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-3 i \, b c^{2} d - 20 i \, a c d^{2} - 9 i \, b d^{3}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 6 \, {\left (3 \, b d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (6 \, b c d^{2} + 5 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{45 \, d^{2} f} \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 \left (a + b \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, 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 = -0.00 \begin {gather*} \int \left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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