Optimal. Leaf size=43 \[ -\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3852}
\begin {gather*} -\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \csc ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 56, normalized size = 1.30 \begin {gather*} -\frac {2 \cot \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 36, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
default | \(\frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
risch | \(\frac {4 i \left (3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{2 i a}-1\right )}{3 b n \left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{2 i a}-1\right )^{3}}\) | \(225\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1332 vs.
\(2 (41) = 82\).
time = 0.30, size = 1332, normalized size = 30.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 71, normalized size = 1.65 \begin {gather*} -\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\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*} \int \frac {\csc ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, 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 [B]
time = 9.23, size = 49, normalized size = 1.14 \begin {gather*} \frac {4\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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