Optimal. Leaf size=256 \[ -\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\left (c^2+5 c d-12 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+5 d) \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a^2 f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.36, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2844, 3056,
2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {(c+5 d) \left (c^2-d^2\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 )}{3 a^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (c^2+5 c d-12 d^2\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 )}{3 a^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2844
Rule 3056
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a \left (2 c^2+7 c d-3 d^2\right )+\frac {1}{2} a (c-7 d) d \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a^2 (11 c-5 d) d^2+\frac {1}{2} a^2 d \left (c^2+5 c d-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 a^4}\\ &=-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\left (c^2+5 c d-12 d^2\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{6 a^2}+\frac {\left ((c+5 d) \left (c^2-d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{6 a^2}\\ &=-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\left (\left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{6 a^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((c+5 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}{6 a^2 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\left (c^2+5 c d-12 d^2\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)}}{3 a^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+5 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}}}{3 a^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.74, size = 310, normalized size = 1.21 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-\left (\left (c^2+5 c d-6 d^2\right ) (c+d \sin (e+f x))\right )+\frac {(c-d) \left (7 d \cos \left (\frac {1}{2} (e+f x)\right )-(c+6 d) \cos \left (\frac {3}{2} (e+f x)\right )+(3 c+11 d) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+d^2 (-11 c+5 d) 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}}+\left (c^2+5 c d-12 d^2\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}}\right )}{3 a^2 f (1+\sin (e+f x))^2 \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. \(1371\) vs.
\(2(302)=604\).
time = 21.12, size = 1372, normalized size = 5.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(1372\) |
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 = 0.19, size = 1185, normalized size = 4.63 \begin {gather*} \frac {{\left (\sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )}\right )} \sin \left (f x + e\right ) - 2 \, \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )}\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 ) + {\left (\sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )}\right )} \sin \left (f x + e\right ) - 2 \, \sqrt {2} {\left (2 \, c^{3} + 10 \, c^{2} d + 9 \, c d^{2} - 15 \, d^{3}\right )}\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 \, {\left (\sqrt {2} {\left (i \, c^{2} d + 5 i \, c d^{2} - 12 i \, d^{3}\right )} \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (-i \, c^{2} d - 5 i \, c d^{2} + 12 i \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (\sqrt {2} {\left (-i \, c^{2} d - 5 i \, c d^{2} + 12 i \, d^{3}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (-i \, c^{2} d - 5 i \, c d^{2} + 12 i \, d^{3}\right )}\right )} \sin \left (f x + e\right ) + 2 \, \sqrt {2} {\left (-i \, c^{2} d - 5 i \, c d^{2} + 12 i \, d^{3}\right )}\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 \, {\left (\sqrt {2} {\left (-i \, c^{2} d - 5 i \, c d^{2} + 12 i \, d^{3}\right )} \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (i \, c^{2} d + 5 i \, c d^{2} - 12 i \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (\sqrt {2} {\left (i \, c^{2} d + 5 i \, c d^{2} - 12 i \, d^{3}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (i \, c^{2} d + 5 i \, c d^{2} - 12 i \, d^{3}\right )}\right )} \sin \left (f x + e\right ) + 2 \, \sqrt {2} {\left (i \, c^{2} d + 5 i \, c d^{2} - 12 i \, d^{3}\right )}\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 (c^{2} d - 2 \, c d^{2} + d^{3} + {\left (c^{2} d + 5 \, c d^{2} - 6 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c^{2} d + 3 \, c d^{2} - 5 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d - 2 \, c d^{2} + d^{3} - {\left (c^{2} d + 5 \, c d^{2} - 6 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{18 \, {\left (a^{2} d f \cos \left (f x + e\right )^{2} - a^{2} d f \cos \left (f x + e\right ) - 2 \, a^{2} d f - {\left (a^{2} d f \cos \left (f x + e\right ) + 2 \, a^{2} d f\right )} \sin \left (f x + e\right )\right )}} \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 = -0.00 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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