Optimal. Leaf size=131 \[ -\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194,
2191, 2188, 29} \begin {gather*} \frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2191
Rule 2194
Rule 2202
Rubi steps
\begin {align*} \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {(3 b) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx}{-b x+\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {(3 b) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {(3 b) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {(3 b) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\left (3 b^2\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 93, normalized size = 0.71 \begin {gather*} -\frac {b^3 x^3-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+2 \coth ^{-1}(\tanh (a+b x))^3+3 b x \coth ^{-1}(\tanh (a+b x))^2 \left (1+2 \log (x)-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x \coth ^{-1}(\tanh (a+b x))^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.89, size = 245, normalized size = 1.87 \begin {gather*} \frac {48 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {48 \, b \log \left (x\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {8 \, {\left (12 \, b^{2} x^{2} - \pi ^{2} - 4 i \, \pi a + 4 \, a^{2} - 9 \, {\left (i \, \pi b - 2 \, a b\right )} x\right )}}{4 \, {\left (i \, \pi ^{3} b^{2} - 6 \, \pi ^{2} a b^{2} - 12 i \, \pi a^{2} b^{2} + 8 \, a^{3} b^{2}\right )} x^{3} + 4 \, {\left (\pi ^{4} b + 8 i \, \pi ^{3} a b - 24 \, \pi ^{2} a^{2} b - 32 i \, \pi a^{3} b + 16 \, a^{4} b\right )} x^{2} - {\left (i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}\right )} x} \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. 1078 vs.
\(2 (129) = 258\).
time = 0.39, size = 1078, normalized size = 8.23 \begin {gather*} \frac {8 \, {\left (6 \, \pi ^{8} a + 64 \, \pi ^{6} a^{3} + 192 \, \pi ^{4} a^{5} - 512 \, a^{9} + 96 \, {\left (3 \, \pi ^{4} a b^{4} + 8 \, \pi ^{2} a^{3} b^{4} - 16 \, a^{5} b^{4}\right )} x^{4} - 12 \, {\left (\pi ^{6} b^{3} - 92 \, \pi ^{4} a^{2} b^{3} - 272 \, \pi ^{2} a^{4} b^{3} + 448 \, a^{6} b^{3}\right )} x^{3} + 8 \, {\left (11 \, \pi ^{6} a b^{2} + 228 \, \pi ^{4} a^{3} b^{2} + 528 \, \pi ^{2} a^{5} b^{2} - 832 \, a^{7} b^{2}\right )} x^{2} - {\left (5 \, \pi ^{8} b - 176 \, \pi ^{6} a^{2} b - 1440 \, \pi ^{4} a^{4} b - 1792 \, \pi ^{2} a^{6} b + 3328 \, a^{8} b\right )} x - 96 \, {\left (16 \, {\left (\pi ^{3} a b^{5} - 4 \, \pi a^{3} b^{5}\right )} x^{5} + 64 \, {\left (\pi ^{3} a^{2} b^{4} - 4 \, \pi a^{4} b^{4}\right )} x^{4} + 8 \, {\left (\pi ^{5} a b^{3} + 8 \, \pi ^{3} a^{3} b^{3} - 48 \, \pi a^{5} b^{3}\right )} x^{3} + 16 \, {\left (\pi ^{5} a^{2} b^{2} - 16 \, \pi a^{6} b^{2}\right )} x^{2} + {\left (\pi ^{7} a b + 4 \, \pi ^{5} a^{3} b - 16 \, \pi ^{3} a^{5} b - 64 \, \pi a^{7} b\right )} x\right )} \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) + 3 \, {\left (16 \, {\left (\pi ^{4} b^{5} - 24 \, \pi ^{2} a^{2} b^{5} + 16 \, a^{4} b^{5}\right )} x^{5} + 64 \, {\left (\pi ^{4} a b^{4} - 24 \, \pi ^{2} a^{3} b^{4} + 16 \, a^{5} b^{4}\right )} x^{4} + 8 \, {\left (\pi ^{6} b^{3} - 12 \, \pi ^{4} a^{2} b^{3} - 272 \, \pi ^{2} a^{4} b^{3} + 192 \, a^{6} b^{3}\right )} x^{3} + 16 \, {\left (\pi ^{6} a b^{2} - 20 \, \pi ^{4} a^{3} b^{2} - 80 \, \pi ^{2} a^{5} b^{2} + 64 \, a^{7} b^{2}\right )} x^{2} + {\left (\pi ^{8} b - 16 \, \pi ^{6} a^{2} b - 160 \, \pi ^{4} a^{4} b - 256 \, \pi ^{2} a^{6} b + 256 \, a^{8} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) - 6 \, {\left (16 \, {\left (\pi ^{4} b^{5} - 24 \, \pi ^{2} a^{2} b^{5} + 16 \, a^{4} b^{5}\right )} x^{5} + 64 \, {\left (\pi ^{4} a b^{4} - 24 \, \pi ^{2} a^{3} b^{4} + 16 \, a^{5} b^{4}\right )} x^{4} + 8 \, {\left (\pi ^{6} b^{3} - 12 \, \pi ^{4} a^{2} b^{3} - 272 \, \pi ^{2} a^{4} b^{3} + 192 \, a^{6} b^{3}\right )} x^{3} + 16 \, {\left (\pi ^{6} a b^{2} - 20 \, \pi ^{4} a^{3} b^{2} - 80 \, \pi ^{2} a^{5} b^{2} + 64 \, a^{7} b^{2}\right )} x^{2} + {\left (\pi ^{8} b - 16 \, \pi ^{6} a^{2} b - 160 \, \pi ^{4} a^{4} b - 256 \, \pi ^{2} a^{6} b + 256 \, a^{8} b\right )} x\right )} \log \left (x\right )\right )}}{16 \, {\left (\pi ^{8} b^{4} + 16 \, \pi ^{6} a^{2} b^{4} + 96 \, \pi ^{4} a^{4} b^{4} + 256 \, \pi ^{2} a^{6} b^{4} + 256 \, a^{8} b^{4}\right )} x^{5} + 64 \, {\left (\pi ^{8} a b^{3} + 16 \, \pi ^{6} a^{3} b^{3} + 96 \, \pi ^{4} a^{5} b^{3} + 256 \, \pi ^{2} a^{7} b^{3} + 256 \, a^{9} b^{3}\right )} x^{4} + 8 \, {\left (\pi ^{10} b^{2} + 28 \, \pi ^{8} a^{2} b^{2} + 288 \, \pi ^{6} a^{4} b^{2} + 1408 \, \pi ^{4} a^{6} b^{2} + 3328 \, \pi ^{2} a^{8} b^{2} + 3072 \, a^{10} b^{2}\right )} x^{3} + 16 \, {\left (\pi ^{10} a b + 20 \, \pi ^{8} a^{3} b + 160 \, \pi ^{6} a^{5} b + 640 \, \pi ^{4} a^{7} b + 1280 \, \pi ^{2} a^{9} b + 1024 \, a^{11} b\right )} x^{2} + {\left (\pi ^{12} + 24 \, \pi ^{10} a^{2} + 240 \, \pi ^{8} a^{4} + 1280 \, \pi ^{6} a^{6} + 3840 \, \pi ^{4} a^{8} + 6144 \, \pi ^{2} a^{10} + 4096 \, a^{12}\right )} x} \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 {1}{x^{2} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 262, normalized size = 2.00 \begin {gather*} \frac {48 i \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{4} + 8 \, \pi ^{3} a - 24 i \, \pi ^{2} a^{2} - 32 \, \pi a^{3} + 16 i \, a^{4}} - \frac {48 i \, b \log \left (x\right )}{i \, \pi ^{4} + 8 \, \pi ^{3} a - 24 i \, \pi ^{2} a^{2} - 32 \, \pi a^{3} + 16 i \, a^{4}} + \frac {16 \, {\left (8 \, b^{2} x + 5 i \, \pi b + 10 \, a b\right )}}{8 i \, \pi ^{3} b^{2} x^{2} + 48 \, \pi ^{2} a b^{2} x^{2} - 96 i \, \pi a^{2} b^{2} x^{2} - 64 \, a^{3} b^{2} x^{2} - 8 \, \pi ^{4} b x + 64 i \, \pi ^{3} a b x + 192 \, \pi ^{2} a^{2} b x - 256 i \, \pi a^{3} b x - 128 \, a^{4} b x - 2 i \, \pi ^{5} - 20 \, \pi ^{4} a + 80 i \, \pi ^{3} a^{2} + 160 \, \pi ^{2} a^{3} - 160 i \, \pi a^{4} - 64 \, a^{5}} + \frac {8}{i \, \pi ^{3} x + 6 \, \pi ^{2} a x - 12 i \, \pi a^{2} x - 8 \, a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.26, size = 1074, normalized size = 8.20 \begin {gather*} \frac {\frac {8}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}-\frac {72\,b\,x}{{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2}+\frac {96\,b^2\,x^2}{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}}{x\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )+x^2\,\left (8\,a\,b-4\,b\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )+4\,b^2\,x^3}+\frac {96\,b\,\mathrm {atanh}\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4+24\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+16\,a^4-8\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-32\,a^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4}-\frac {4\,b\,x\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3}\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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