Optimal. Leaf size=103 \[ \frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}+e^{2 c (a+b x)}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}+e^{2 c (a+b x)}\right )}{2 b c} \]
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Rubi [A]
time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2225, 5316,
2320, 12, 1261, 646, 31} \begin {gather*} \frac {\left (1-\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt {2}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt {2}\right )}{2 b c}+\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 646
Rule 1261
Rule 2225
Rule 2320
Rule 5316
Rubi steps
\begin {align*} \int e^{c (a+b x)} \cot ^{-1}(\cosh (a c+b c x)) \, dx &=\frac {\text {Subst}\left (\int e^x \cot ^{-1}(\cosh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int \frac {e^x \sinh (x)}{1+\cosh ^2(x)} \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int \frac {2 x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int \frac {-1+x}{1+6 x+x^2} \, dx,x,e^{2 a c+2 b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3-2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3+2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}\\ &=\frac {e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}+e^{2 a c+2 b c x}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}+e^{2 a c+2 b c x}\right )}{2 b c}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.08, size = 146, normalized size = 1.42 \begin {gather*} \frac {4 c (a+b x)+2 e^{c (a+b x)} \cot ^{-1}\left (\frac {1}{2} e^{-c (a+b x)} \left (1+e^{2 c (a+b x)}\right )\right )+\text {RootSum}\left [1+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-a c-b c x+\log \left (e^{c (a+b x)}-\text {$\#$1}\right )-7 a c \text {$\#$1}^2-7 b c x \text {$\#$1}^2+7 \log \left (e^{c (a+b x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+3 \text {$\#$1}^2}\&\right ]}{2 b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.27, size = 1358, normalized size = 13.18
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1358\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 131, normalized size = 1.27 \begin {gather*} \frac {\operatorname {arccot}\left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, b c x - 2 \, a c\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, b c x - 2 \, a c\right )} + 3}\right )}{2 \, b c} + \frac {2 \, {\left (b c x + a c\right )}}{b c} + \frac {\log \left (6 \, e^{\left (-2 \, b c x - 2 \, a c\right )} + e^{\left (-4 \, b c x - 4 \, a c\right )} + 1\right )}{2 \, b c} \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. 276 vs.
\(2 (86) = 172\).
time = 2.33, size = 276, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )}}{\cosh \left (b c x + a c\right )^{2} + 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2} + 1}\right ) + \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \sqrt {2} + 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3}\right ) + \log \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \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*} e^{a c} \int e^{b c x} \operatorname {acot}{\left (\cosh {\left (a c + b c x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 154, normalized size = 1.50 \begin {gather*} -\frac {{\left (\sqrt {2} e^{\left (-a c\right )} \log \left (-\frac {2 \, \sqrt {2} e^{\left (2 \, a c\right )} - e^{\left (2 \, b c x + 4 \, a c\right )} - 3 \, e^{\left (2 \, a c\right )}}{2 \, \sqrt {2} e^{\left (2 \, a c\right )} + e^{\left (2 \, b c x + 4 \, a c\right )} + 3 \, e^{\left (2 \, a c\right )}}\right ) - 2 \, \arctan \left (\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}}\right ) e^{\left (b c x\right )} - e^{\left (-a c\right )} \log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{2 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 133, normalized size = 1.29 \begin {gather*} \frac {\ln \left (8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-2\,\sqrt {2}-6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}+1\right )}{2\,b\,c}-\frac {\ln \left (8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}+2\,\sqrt {2}+6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}-1\right )}{2\,b\,c}+\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}\right )}{b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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