Optimal. Leaf size=80 \[ -\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 80, 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 = {2866, 2864,
129, 440} \begin {gather*} -\frac {\cos (c+d x) F_1\left (\frac {1}{2};1,\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}} \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 (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {\csc (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) \text {Subst}\left (\int \frac {1}{(1-x) (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)) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \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};1,\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 [F]
time = 6.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\csc \left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]
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 \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 {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, 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 )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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