Optimal. Leaf size=82 \[ -\frac {\csc ^4(c+d x)}{4 a d}-\frac {\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\cot (c+d x) \csc (c+d x)}{8 a d} \]
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Rubi [A] time = 0.16, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2835, 2606, 30, 2611, 3768, 3770} \[ -\frac {\csc ^4(c+d x)}{4 a d}-\frac {\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\cot (c+d x) \csc (c+d x)}{8 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^2(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\int \csc ^3(c+d x) \, dx}{4 a}-\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\int \csc (c+d x) \, dx}{8 a}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\csc ^4(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 91, normalized size = 1.11 \[ -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{16 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 138, normalized size = 1.68 \[ \frac {2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (d x + c\right ) + 4}{16 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 129, normalized size = 1.57 \[ -\frac {\frac {2 \, {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {2 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} - \frac {\frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 72, normalized size = 0.88 \[ \frac {1}{8 a d \left (-1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{16 a d}-\frac {1}{8 a d \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{16 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 86, normalized size = 1.05 \[ \frac {\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 2\right )}}{a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 75, normalized size = 0.91 \[ -\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{8\,a\,d}-\frac {\frac {{\cos \left (c+d\,x\right )}^2}{8}+\frac {\cos \left (c+d\,x\right )}{8}+\frac {1}{4}}{d\,\left (-a\,{\cos \left (c+d\,x\right )}^3-a\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+a\right )} \]
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 \[ \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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