Optimal. Leaf size=161 \[ \frac {37 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt [3]{a \sin (c+d x)+a}}+\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a \sin (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2783, 2968, 3023, 2751, 2652, 2651} \[ \frac {37 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt [3]{a \sin (c+d x)+a}}+\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2651
Rule 2652
Rule 2751
Rule 2783
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx &=-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \int \frac {\sin (c+d x) \left (2 a-\frac {1}{3} a \sin (c+d x)\right )}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{8 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \int \frac {2 a \sin (c+d x)-\frac {1}{3} a \sin ^2(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{8 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}+\frac {9 \int \frac {-\frac {2 a^2}{9}+\frac {11}{3} a^2 \sin (c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{40 a^2}\\ &=-\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}-\frac {37}{80} \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx\\ &=-\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}-\frac {\left (37 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac {1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{80 \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {37 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.46, size = 110, normalized size = 0.68 \[ \frac {3 \cos (c+d x) \left (\sqrt {1-\sin (c+d x)} (2 \sin (c+d x)+5 \cos (2 (c+d x))-36)-37 \sqrt {2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )\right )}{80 d \sqrt {1-\sin (c+d x)} \sqrt [3]{a (\sin (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________