Optimal. Leaf size=77 \[ -\frac {\sqrt [6]{2} F_1\left (\frac {1}{2};2,\frac {5}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2866, 2864,
129, 440} \begin {gather*} -\frac {\sqrt [6]{2} \cos (c+d x) F_1\left (\frac {1}{2};2,\frac {5}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 440
Rule 2864
Rule 2866
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx &=\frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {\csc ^2(c+d x)}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{\sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {\cos (c+d x) \text {Subst}\left (\int \frac {1}{(1-x)^2 (2-x)^{5/6} \sqrt {x}} \, dx,x,1-\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {(2 \cos (c+d x)) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (2-x^2\right )^{5/6}} \, dx,x,\sqrt {1-\sin (c+d x)}\right )}{d \sqrt {1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {\sqrt [6]{2} F_1\left (\frac {1}{2};2,\frac {5}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.49, size = 184, normalized size = 2.39 \begin {gather*} \frac {2\ 2^{2/3} \cos ^{\frac {2}{3}}\left (\frac {1}{4} (2 c-\pi +2 d x)\right ) (\cos (c+d x)+i \sin (c+d x)) \left (1+4 \sin (c+d x)+4 i \cos (c+d x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-i e^{-i (c+d x)}\right ) (1+i \cos (c+d x)+\sin (c+d x))^{2/3}\right )}{5 d \left (-(-1)^{3/4} e^{-\frac {1}{2} i (c+d x)} \left (i+e^{i (c+d x)}\right )\right )^{2/3} \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{a (1+\sin (c+d x))}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{2}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{2}{\left (c + d x \right )}}{\sqrt [3]{a \left (\sin {\left (c + d x \right )} + 1\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 = -0.01 \begin {gather*} \int \frac {1}{{\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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