Optimal. Leaf size=77 \[ -\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}+\frac {x \left (a^2+2 b^2\right )}{2 a^3}+\frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3} \]
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Rubi [A] time = 0.21, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3872, 2865, 2735, 2660, 618, 204} \[ \frac {x \left (a^2+2 b^2\right )}{2 a^3}+\frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3}-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2865
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+b \text {csch}(x)} \, dx &=i \int \frac {\cosh ^2(x) \sinh (x)}{i b+i a \sinh (x)} \, dx\\ &=-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}+\frac {\int \frac {-i a b+i \left (a^2+2 b^2\right ) \sinh (x)}{i b+i a \sinh (x)} \, dx}{2 a^2}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}-\frac {\left (i b \left (a^2+b^2\right )\right ) \int \frac {1}{i b+i a \sinh (x)} \, dx}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}-\frac {\left (2 i b \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{i b+2 i a x-i b x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}+\frac {\left (4 i b \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,2 i a-2 i b \tanh \left (\frac {x}{2}\right )\right )}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}+\frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3}-\frac {\cosh (x) (2 b-a \sinh (x))}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 80, normalized size = 1.04 \[ \frac {8 b \sqrt {-a^2-b^2} \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+2 a^2 x+a^2 \sinh (2 x)-4 a b \cosh (x)+4 b^2 x}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 304, normalized size = 3.95 \[ \frac {a^{2} \cosh \relax (x)^{4} + a^{2} \sinh \relax (x)^{4} - 4 \, a b \cosh \relax (x)^{3} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} x \cosh \relax (x)^{2} + 4 \, {\left (a^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x)^{3} - 4 \, a b \cosh \relax (x) + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} - 6 \, a b \cosh \relax (x) + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} x\right )} \sinh \relax (x)^{2} + 8 \, {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) - a}\right ) - a^{2} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} - 3 \, a b \cosh \relax (x)^{2} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} x \cosh \relax (x) - a b\right )} \sinh \relax (x)}{8 \, {\left (a^{3} \cosh \relax (x)^{2} + 2 \, a^{3} \cosh \relax (x) \sinh \relax (x) + a^{3} \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 121, normalized size = 1.57 \[ \frac {a e^{\left (2 \, x\right )} - 4 \, b e^{x}}{8 \, a^{2}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} x}{2 \, a^{3}} - \frac {{\left (4 \, a b e^{x} + a^{2}\right )} e^{\left (-2 \, x\right )}}{8 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 172, normalized size = 2.23 \[ \frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b^{2}}{a^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b^{2}}{a^{3}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 122, normalized size = 1.58 \[ -\frac {{\left (4 \, b e^{\left (-x\right )} - a\right )} e^{\left (2 \, x\right )}}{8 \, a^{2}} - \frac {4 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )}}{8 \, a^{2}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} x}{2 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 159, normalized size = 2.06 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {b\,{\mathrm {e}}^x}{2\,a^2}-\frac {b\,{\mathrm {e}}^{-x}}{2\,a^2}+\frac {x\,\left (a^2+2\,b^2\right )}{2\,a^3}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x\,\left (a^2+b^2\right )}{a^4}-\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )\,\sqrt {a^2+b^2}}{a^4}\right )\,\sqrt {a^2+b^2}}{a^3}+\frac {b\,\ln \left (\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )\,\sqrt {a^2+b^2}}{a^4}+\frac {2\,b\,{\mathrm {e}}^x\,\left (a^2+b^2\right )}{a^4}\right )\,\sqrt {a^2+b^2}}{a^3} \]
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 \[ \int \frac {\cosh ^{2}{\relax (x )}}{a + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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