Optimal. Leaf size=67 \[ -\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 3556}
\begin {gather*} \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rubi steps
\begin {align*} \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \tan ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\text {Subst}\left (\int \tan ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\text {Subst}\left (\int \tan (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 55, normalized size = 0.82 \begin {gather*} -\frac {4 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )+2 \tan ^2\left (a+b \log \left (c x^n\right )\right )-\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 57, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4}-\frac {\left (\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\) | \(57\) |
default | \(\frac {\frac {\left (\tan ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4}-\frac {\left (\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\) | \(57\) |
risch | \(-i \ln \left (x \right )+\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 {2 i \ln \left (c \right )}{n}+\frac {2 i \ln \left (x^{n}\right )}{n}+\frac {2 i a}{n b}-\frac {4 \left (x^{n}\right )^{2 i b} c^{2 i b} \left (c^{4 i b} \left (x^{n}\right )^{4 i b} {\mathrm e}^{3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{3 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}^{6 i a}+c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{2 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-2 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{2 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}^{4 i a}+{\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}\right )}{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 )^{4}}-\frac {\ln \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 )}{b n}\) | \(667\) |
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. 4466 vs.
\(2 (63) = 126\).
time = 0.38, size = 4466, normalized size = 66.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (63) = 126\).
time = 3.30, size = 129, normalized size = 1.93 \begin {gather*} -\frac {{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac {1}{2}\right ) + 4 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.76, size = 82, normalized size = 1.22 \begin {gather*} \begin {cases} \log {\left (x \right )} \tan ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tan ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} + \frac {\tan ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b n} - \frac {\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b 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 = 6.59, size = 247, normalized size = 3.69 \begin {gather*} \ln \left (x\right )\,1{}\mathrm {i}+\frac {8}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {4}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}+\frac {4}{b\,n\,\left (4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+4\,{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,8{}\mathrm {i}}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}{b\,n}-\frac {8}{b\,n\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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