Optimal. Leaf size=49 \[ -\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac {3 \cosh (a+b x)}{2 b}-\frac {\cosh (a+b x) \coth ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2672, 294, 327,
212} \begin {gather*} \frac {3 \cosh (a+b x)}{2 b}-\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac {\cosh (a+b x) \coth ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 294
Rule 327
Rule 2672
Rubi steps
\begin {align*} \int \cosh (a+b x) \coth ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\cosh (a+b x) \coth ^2(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=\frac {3 \cosh (a+b x)}{2 b}-\frac {\cosh (a+b x) \coth ^2(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac {3 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac {3 \cosh (a+b x)}{2 b}-\frac {\cosh (a+b x) \coth ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 1.37 \begin {gather*} \frac {\cosh (a+b x)}{b}-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {3 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \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. \(89\) vs.
\(2(43)=86\).
time = 1.46, size = 90, normalized size = 1.84
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}\) | \(90\) |
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. 108 vs.
\(2 (43) = 86\).
time = 0.28, size = 108, normalized size = 2.20 \begin {gather*} \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} - \frac {4 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - 1}{2 \, b {\left (e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\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. 612 vs.
\(2 (43) = 86\).
time = 0.36, size = 612, normalized size = 12.49 \begin {gather*} \frac {\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 3 \, \cosh \left (b x + a\right )^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{3} - 2 \, \cosh \left (b x + a\right )^{3} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (5 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{3} - 2 \, \cosh \left (b x + a\right )^{3} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (5 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \, {\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} - 2 \, b \cosh \left (b x + a\right )^{3} + 2 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + b \cosh \left (b x + a\right ) + {\left (5 \, b \cosh \left (b x + a\right )^{4} - 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )\right )}} \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 \cosh {\left (a + b x \right )} \coth ^{3}{\left (a + b x \right )}\, dx \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. 105 vs.
\(2 (43) = 86\).
time = 0.40, size = 105, normalized size = 2.14 \begin {gather*} -\frac {\frac {4 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4} - 2 \, e^{\left (b x + a\right )} - 2 \, e^{\left (-b x - a\right )} + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 112, normalized size = 2.29 \begin {gather*} \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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