Optimal. Leaf size=26 \[ -\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3556}
\begin {gather*} -\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \tan (d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.96 \begin {gather*} -\frac {\log \left (\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 30, normalized size = 1.15
method | result | size |
derivativedivides | \(\frac {\ln \left (1+\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{2 n b d}\) | \(30\) |
default | \(\frac {\ln \left (1+\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{2 n b d}\) | \(30\) |
risch | \(-i \ln \left (x \right )+\frac {2 i a}{n b}+\frac {2 i \ln \left (c \right )}{n}+\frac {2 i \ln \left (x^{n}\right )}{n}+\frac {\pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {\pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{n}-\frac {\pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{n}+\frac {\pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{n}-\frac {\ln \left ({\mathrm e}^{i d \left (-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )+2 b \ln \left (c \right )+2 b \ln \left (x^{n}\right )+2 a \right )}+1\right )}{b d n}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.92 \begin {gather*} \frac {\log \left (\sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.80, size = 35, normalized size = 1.35 \begin {gather*} -\frac {\log \left (\frac {1}{2} \, \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + \frac {1}{2}\right )}{2 \, b d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.37, size = 44, normalized size = 1.69 \begin {gather*} \begin {cases} \log {\left (x \right )} \tan {\left (a d \right )} & \text {for}\: b = 0 \\0 & \text {for}\: d = 0 \\\log {\left (x \right )} \tan {\left (a d + b d \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\cos {\left (a d + b d \log {\left (c x^{n} \right )} \right )} \right )}}{b d n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [B]
time = 3.78, size = 38, normalized size = 1.46 \begin {gather*} \ln \left (x\right )\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}+1\right )}{b\,d\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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