Optimal. Leaf size=74 \[ -\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {d^2 \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4269, 3797,
2221, 2317, 2438} \begin {gather*} \frac {d^2 \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {(c+d x)^2}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 4269
Rubi steps
\begin {align*} \int (c+d x)^2 \text {csch}^2(a+b x) \, dx &=-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {(2 d) \int (c+d x) \coth (a+b x) \, dx}{b}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {(4 d) \int \frac {e^{2 (a+b x)} (c+d x)}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {\left (2 d^2\right ) \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {d^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {d^2 \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.68, size = 198, normalized size = 2.68 \begin {gather*} \frac {\text {csch}(a) \left (-2 b c d (b x \cosh (a)-\log (\sinh (a+b x)) \sinh (a))+d^2 \left (-b^2 e^{-\tanh ^{-1}(\tanh (a))} x^2 \cosh (a) \sqrt {\text {sech}^2(a)}+i b \pi x \sinh (a)-i \pi \log \left (1+e^{2 b x}\right ) \sinh (a)+2 b x \log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right ) \sinh (a)+i \pi \log (\cosh (b x)) \sinh (a)+2 \tanh ^{-1}(\tanh (a)) \left (b x+\log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )-\log \left (i \sinh \left (b x+\tanh ^{-1}(\tanh (a))\right )\right )\right ) \sinh (a)-\text {PolyLog}\left (2,e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right ) \sinh (a)\right )+b^2 (c+d x)^2 \text {csch}(a+b x) \sinh (b x)\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(239\) vs.
\(2(74)=148\).
time = 0.83, size = 240, normalized size = 3.24
method | result | size |
risch | \(-\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 d c \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d^{2} x^{2}}{b}-\frac {4 d^{2} a x}{b^{2}}-\frac {2 d^{2} a^{2}}{b^{3}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}+\frac {2 d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {4 d^{2} a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 623 vs.
\(2 (73) = 146\).
time = 0.41, size = 623, normalized size = 8.42 \begin {gather*} -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \sinh \left (b x + a\right )^{2} - {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} - d^{2}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} - d^{2}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b d^{2} x + b c d - {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b c d - a d^{2} - {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b d^{2} x + a d^{2} - {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}} \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 \left (c + d x\right )^{2} \operatorname {csch}^{2}{\left (a + b x \right )}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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