Optimal. Leaf size=375 \[ \frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.44, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2872, 3102,
2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {2 b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \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 )}{105 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c 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 )}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2872
Rule 3102
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int \sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} \left (2 b^3 c+7 a^3 d+3 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-21 a^2 d-5 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-4 a d) \sin ^2(e+f x)\right ) \, dx}{7 d}\\ &=\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {1}{4} d \left (2 b^3 c-35 a^3 d-63 a b^2 d\right )-\frac {1}{4} b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{35 d^2}\\ &=\frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {8 \int \frac {\frac {1}{8} d \left (105 a^3 c d+147 a b^2 c d+105 a^2 b d^2+b^3 \left (2 c^2+25 d^2\right )\right )+\frac {1}{8} \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^2}\\ &=\frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^3}+\frac {\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{105 d^3}\\ &=\frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c 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}{105 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\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}{105 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c 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)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\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}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.51, size = 306, normalized size = 0.82 \begin {gather*} \frac {-4 \left (d^2 \left (105 a^3 c d+147 a b^2 c d+105 a^2 b d^2+b^3 \left (2 c^2+25 d^2\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+\left (105 a^2 b c d^2+105 a^3 d^3+21 a b^2 d \left (-2 c^2+9 d^2\right )+b^3 \left (8 c^3+19 c 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 )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+b d (c+d \sin (e+f x)) \left (\left (-84 a b c d-420 a^2 d^2+b^2 \left (16 c^2-115 d^2\right )\right ) \cos (e+f x)+3 b d (5 b d \cos (3 (e+f x))-2 (b c+21 a d) \sin (2 (e+f x)))\right )}{210 d^3 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. \(1560\) vs.
\(2(417)=834\).
time = 26.81, size = 1561, normalized size = 4.16
method | result | size |
default | \(\text {Expression too large to display}\) | \(1561\) |
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 = 0.20, size = 743, normalized size = 1.98 \begin {gather*} -\frac {\sqrt {2} {\left (16 \, b^{3} c^{4} - 84 \, a b^{2} c^{3} d + 2 \, {\left (105 \, a^{2} b + 16 \, b^{3}\right )} c^{2} d^{2} - 21 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c d^{3} - 15 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} d^{4}\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 (16 \, b^{3} c^{4} - 84 \, a b^{2} c^{3} d + 2 \, {\left (105 \, a^{2} b + 16 \, b^{3}\right )} c^{2} d^{2} - 21 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c d^{3} - 15 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} d^{4}\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 (8 i \, b^{3} c^{3} d - 42 i \, a b^{2} c^{2} d^{2} + i \, {\left (105 \, a^{2} b + 19 \, b^{3}\right )} c d^{3} + 21 i \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} d^{4}\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 (-8 i \, b^{3} c^{3} d + 42 i \, a b^{2} c^{2} d^{2} - i \, {\left (105 \, a^{2} b + 19 \, b^{3}\right )} c d^{3} - 21 i \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} d^{4}\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 (15 \, b^{3} d^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 5 \, {\left (21 \, a^{2} b + 8 \, b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{4} 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 )^{3} \sqrt {c + d \sin {\left (e + f 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 = -0.00 \begin {gather*} \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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