Optimal. Leaf size=318 \[ -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 \left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2763
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int (a+a \sin (e+f x)) \left (a^2 (c+5 d)-2 a^2 (c-4 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx}{7 d}\\ &=-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int \sqrt {c+d \sin (e+f x)} \left (a^3 (c+5 d)+\left (-2 a^3 (c-4 d)+a^3 (c+5 d)\right ) \sin (e+f x)-2 a^3 (c-4 d) \sin ^2(e+f x)\right ) \, dx}{7 d}\\ &=\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a^3 (c-49 d) d+\frac {1}{2} a^3 \left (4 c^2-21 c d+65 d^2\right ) \sin (e+f x)\right ) \, dx}{35 d^2}\\ &=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {8 \int \frac {\frac {1}{4} a^3 d \left (c^2+126 c d+65 d^2\right )+\frac {1}{4} a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^2}\\ &=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^3}+\frac {\left (2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{105 d^3}\\ &=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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 (2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 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}{105 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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 {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 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}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.72, size = 266, normalized size = 0.84 \[ -\frac {a^3 \left (-2 d \cos (e+f x) \left (16 c^3+d \left (4 c^2-336 c d-565 d^2\right ) \sin (e+f x)-84 c^2 d+18 d^2 (2 c+7 d) \cos (2 (e+f x))-556 c d^2+15 d^3 \sin (3 (e+f x))-126 d^3\right )-16 \left (4 c^4-21 c^3 d+61 c^2 d^2+21 c d^3-65 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+16 \left (4 c^4-17 c^3 d+41 c^2 d^2+209 c d^3+147 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )}{420 d^3 f \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.38, size = 1316, normalized size = 4.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________