Optimal. Leaf size=77 \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]
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Rubi [A] time = 0.13149, antiderivative size = 77, normalized size of antiderivative = 1., 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 = {2787, 2785, 130, 429} \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]
Antiderivative was successfully verified.
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Rule 2787
Rule 2785
Rule 130
Rule 429
Rubi steps
\begin{align*} \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=\frac{(a+a \sin (c+d x))^{2/3} \int \csc ^2(c+d x) (1+\sin (c+d x))^{2/3} \, dx}{(1+\sin (c+d x))^{2/3}}\\ &=-\frac{\left (\cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{2-x}}{(1-x)^2 \sqrt{x}} \, dx,x,1-\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}}\\ &=-\frac{\left (2 \cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{2-x^2}}{\left (1-x^2\right )^2} \, dx,x,\sqrt{1-\sin (c+d x)}\right )}{d \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}}\\ &=-\frac{2 \sqrt [6]{2} F_1\left (\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right ) \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{d (1+\sin (c+d x))^{7/6}}\\ \end{align*}
Mathematica [C] time = 14.6154, size = 143, normalized size = 1.86 \[ -\frac{2 e^{i (c+d x)} \left (\left (1+i e^{-i (c+d x)}\right )^{2/3} \left (e^{i (c+d x)}-i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right )-e^{i (c+d x)}-i\right ) (a (\sin (c+d x)+1))^{2/3}}{d \left (e^{i (c+d x)}-i\right ) \left (e^{i (c+d x)}+i\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \csc \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \csc \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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