Optimal. Leaf size=82 \[ \frac {2 \cot (c+d x)}{a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cot (c+d x) \csc (c+d x)}{d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2768, 2748, 3768, 3770, 3767, 8} \[ \frac {2 \cot (c+d x)}{a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cot (c+d x) \csc (c+d x)}{d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2768
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\cot (c+d x) \csc (c+d x)}{d (a+a \sin (c+d x))}-\frac {\int \csc ^3(c+d x) (-3 a+2 a \sin (c+d x)) \, dx}{a^2}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{d (a+a \sin (c+d x))}-\frac {2 \int \csc ^2(c+d x) \, dx}{a}+\frac {3 \int \csc ^3(c+d x) \, dx}{a}\\ &=-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cot (c+d x) \csc (c+d x)}{d (a+a \sin (c+d x))}+\frac {3 \int \csc (c+d x) \, dx}{2 a}+\frac {2 \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a d}\\ &=-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {2 \cot (c+d x)}{a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cot (c+d x) \csc (c+d x)}{d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 85, normalized size = 1.04 \[ -\frac {4 \tan (c+d x)-4 \csc (2 (c+d x))-3 \sec (c+d x)+\csc ^2(c+d x) \sec (c+d x)+3 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \tanh ^{-1}\left (\sqrt {\cos ^2(c+d x)}\right )}{2 a d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 232, normalized size = 2.83 \[ \frac {8 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right ) - 4}{4 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d + {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 112, normalized size = 1.37 \[ \frac {\frac {12 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} + \frac {16}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 115, normalized size = 1.40 \[ \frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {1}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{2 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}+\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 157, normalized size = 1.91 \[ -\frac {\frac {\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}{\frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 116, normalized size = 1.41 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d}+\frac {10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{2}}{d\,\left (4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]
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 \[ \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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