Optimal. Leaf size=82 \[ -\frac {\tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\csc ^4(c+d x)}{4 a d} \]
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Rubi [A]
time = 0.12, 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 = {3957, 2914,
2686, 30, 2691, 3853, 3855} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rule 3957
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 {\text {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 = 0.27, size = 91, normalized size = 1.11 \begin {gather*} -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x)}{16 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 55, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {1}{-8+8 \cos \left (d x +c \right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{8 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d a}\) | \(55\) |
default | \(\frac {\frac {1}{-8+8 \cos \left (d x +c \right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{8 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d a}\) | \(55\) |
norman | \(\frac {-\frac {1}{16 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}\) | \(79\) |
risch | \(\frac {{\mathrm e}^{5 i \left (d x +c \right )}+2 \,{\mathrm e}^{4 i \left (d x +c \right )}+10 \,{\mathrm e}^{3 i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}{4 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 86, normalized size = 1.05 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.68, size = 138, normalized size = 1.68 \begin {gather*} \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 )}} \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*} \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 129, normalized size = 1.57 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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