Optimal. Leaf size=80 \[ -\frac {\cos (c+d x) F_1\left (\frac {1}{2};2,\frac {11}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.14, antiderivative size = 80, 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 = {2787, 2785, 130, 429} \[ -\frac {\cos (c+d x) F_1\left (\frac {1}{2};2,\frac {11}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 2785
Rule 2787
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {\csc ^2(c+d x)}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {\cos (c+d x) \operatorname {Subst}\left (\int \frac {1}{(1-x)^2 (2-x)^{11/6} \sqrt {x}} \, dx,x,1-\sin (c+d x)\right )}{a 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)) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (2-x^2\right )^{11/6}} \, dx,x,\sqrt {1-\sin (c+d x)}\right )}{a d \sqrt {1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {F_1\left (\frac {1}{2};2,\frac {11}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 14.18, size = 230, normalized size = 2.88 \[ \frac {8\ 2^{2/3} \cos ^{\frac {8}{3}}\left (\frac {1}{4} (2 c+2 d x-\pi )\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left (14 i \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-i e^{-i (c+d x)}\right ) (\sin (c+d x)+i \cos (c+d x)+1)^{2/3} (\sin (2 (c+d x))+2 \cos (c+d x))+35 \sin (c+d x)-14 \cos (2 (c+d x))+6\right )}{55 d \left (-1+i e^{i (c+d x)}\right )^3 \left (e^{i (c+d x)}-i\right ) \left (-(-1)^{3/4} e^{-\frac {1}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right )\right )^{2/3} (a (\sin (c+d x)+1))^{4/3}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {\csc \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {\csc ^{2}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {\csc \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\csc ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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