Optimal. Leaf size=202 \[ -\frac {64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f} \]
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Rubi [A]
time = 0.19, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2840, 2830,
2726, 2725} \begin {gather*} -\frac {64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 f}-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac {4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2726
Rule 2830
Rule 2840
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx &=-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {2 \int (a+a \sin (e+f x))^{5/2} \left (\frac {1}{2} a \left (9 c^2+7 d^2\right )+a (9 c-d) d \sin (e+f x)\right ) \, dx}{9 a}\\ &=-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{21} \left (21 c^2+30 c d+13 d^2\right ) \int (a+a \sin (e+f x))^{5/2} \, dx\\ &=-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{105} \left (8 a \left (21 c^2+30 c d+13 d^2\right )\right ) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac {16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac {1}{315} \left (32 a^2 \left (21 c^2+30 c d+13 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}\\ \end {align*}
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Mathematica [A]
time = 2.11, size = 180, normalized size = 0.89 \begin {gather*} -\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (7476 c^2+12480 c d+5653 d^2-4 \left (63 c^2+360 c d+254 d^2\right ) \cos (2 (e+f x))+35 d^2 \cos (4 (e+f x))+2352 c^2 \sin (e+f x)+6060 c d \sin (e+f x)+3116 d^2 \sin (e+f x)-180 c d \sin (3 (e+f x))-260 d^2 \sin (3 (e+f x))\right )}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.09, size = 168, normalized size = 0.83
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (35 d^{2} \left (\sin ^{4}\left (f x +e \right )\right )+90 c d \left (\sin ^{3}\left (f x +e \right )\right )+130 d^{2} \left (\sin ^{3}\left (f x +e \right )\right )+63 c^{2} \left (\sin ^{2}\left (f x +e \right )\right )+360 c d \left (\sin ^{2}\left (f x +e \right )\right )+219 d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+294 c^{2} \sin \left (f x +e \right )+690 c d \sin \left (f x +e \right )+292 d^{2} \sin \left (f x +e \right )+903 c^{2}+1380 c d +584 d^{2}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(168\) |
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 [A]
time = 0.34, size = 352, normalized size = 1.74 \begin {gather*} -\frac {2 \, {\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{5} - 5 \, {\left (18 \, a^{2} c d + 19 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} - {\left (63 \, a^{2} c^{2} + 360 \, a^{2} c d + 289 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (231 \, a^{2} c^{2} + 510 \, a^{2} c d + 263 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (483 \, a^{2} c^{2} + 870 \, a^{2} c d + 419 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} + 10 \, {\left (9 \, a^{2} c d + 13 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, a^{2} c^{2} + 90 \, a^{2} c d + 53 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (147 \, a^{2} c^{2} + 390 \, a^{2} c d + 211 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \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 \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 348, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} {\left (35 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) + 630 \, {\left (20 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 30 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 13 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 420 \, {\left (5 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 252 \, {\left (a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 45 \, {\left (4 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right )\right )} \sqrt {a}}{2520 \, f} \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+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\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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