Optimal. Leaf size=77 \[ -\frac {2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac {1}{2};1,-\frac {1}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]
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Rubi [A] time = 0.11, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2787, 2785, 130, 429} \[ -\frac {2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac {1}{2};1,-\frac {1}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 2785
Rule 2787
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=\frac {(a+a \sin (c+d x))^{2/3} \int \csc (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) \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}}{1-x^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};1,-\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*}
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Mathematica [F] time = 2.79, size = 0, normalized size = 0.00 \[ \int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \csc \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \csc \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}\, dx \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \csc \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3}}{\sin \left (c+d\,x\right )} \,d x \]
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 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \csc {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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