Optimal. Leaf size=66 \[ \frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
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Rubi [A]
time = 0.04, antiderivative size = 66, 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 {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cosh \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 {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \tanh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\text {Subst}\left (\int \tanh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\text {Subst}\left (\int \tanh (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 55, normalized size = 0.83 \begin {gather*} \frac {4 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )-2 \tanh ^2\left (a+b \log \left (c x^n\right )\right )-\tanh ^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 = 2.28, size = 71, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tanh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4}-\frac {\left (\tanh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(71\) |
default | \(\frac {-\frac {\left (\tanh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4}-\frac {\left (\tanh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(71\) |
risch | \(\ln \left (x \right )-\frac {2 a}{n b}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}+\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{n}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{n}+\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{n}+\frac {4 \left (x^{n}\right )^{2 b} c^{2 b} \left ({\mathrm e}^{6 a} {\mathrm e}^{-3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-3 i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )} \left (x^{n}\right )^{4 b} c^{4 b}+{\mathrm e}^{4 a} {\mathrm e}^{-2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-2 i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )} \left (x^{n}\right )^{2 b} c^{2 b}+{\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}\right )}{b n \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}+1\right )^{4}}+\frac {\ln \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}+1\right )}{b n}\) | \(657\) |
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. 829 vs.
\(2 (62) = 124\).
time = 0.37, size = 829, normalized size = 12.56 \begin {gather*} \frac {48 \, c^{6 \, b} e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 108 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 88 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 25}{24 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {12 \, c^{6 \, b} e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 42 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 52 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 25}{24 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {5 \, {\left (4 \, c^{6 \, b} e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{8 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {5 \, {\left (6 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{12 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {5 \, {\left (4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{12 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {5}{8 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}}{c^{2 \, b}}\right )}{b n} - \log \left (x\right ) \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. 1568 vs.
\(2 (62) = 124\).
time = 0.38, size = 1568, normalized size = 23.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.91, size = 92, normalized size = 1.39 \begin {gather*} \begin {cases} \log {\left (x \right )} \tanh ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \tanh ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} \tanh ^{5}{\left (a \right )} & \text {for}\: b = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\log {\left (\tanh {\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b n} - \frac {\tanh ^{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 [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs.
\(2 (62) = 124\).
time = 0.49, size = 161, normalized size = 2.44 \begin {gather*} \frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} + 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b n} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 227, normalized size = 3.44 \begin {gather*} \frac {8}{b\,n+3\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+3\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}}-\ln \left (x\right )+\frac {4}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\frac {4}{b\,n+4\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+6\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+4\,b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}+b\,n\,{\mathrm {e}}^{8\,a}\,{\left (c\,x^n\right )}^{8\,b}}-\frac {8}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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