Optimal. Leaf size=47 \[ \frac {2 a \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}+\frac {x}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2735, 2660, 618, 206} \[ \frac {2 a \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}+\frac {x}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx &=\frac {x}{b}-\frac {a \int \frac {1}{a+b \sinh (x)} \, dx}{b}\\ &=\frac {x}{b}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b}\\ &=\frac {x}{b}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b}\\ &=\frac {x}{b}+\frac {2 a \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 1.11 \[ \frac {x-\frac {2 a \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 134, normalized size = 2.85 \[ \frac {\sqrt {a^{2} + b^{2}} a \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) + {\left (a^{2} + b^{2}\right )} x}{a^{2} b + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 67, normalized size = 1.43 \[ -\frac {a \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 1.34 \[ -\frac {2 a \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 65, normalized size = 1.38 \[ -\frac {a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 99, normalized size = 2.11 \[ \frac {x}{b}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^2}-\frac {2\,a\,\left (b-a\,{\mathrm {e}}^x\right )}{b^2\,\sqrt {a^2+b^2}}\right )}{b\,\sqrt {a^2+b^2}}+\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^2}+\frac {2\,a\,\left (b-a\,{\mathrm {e}}^x\right )}{b^2\,\sqrt {a^2+b^2}}\right )}{b\,\sqrt {a^2+b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.30, size = 252, normalized size = 5.36 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {\cosh {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {x}{b} & \text {for}\: a = 0 \\\frac {b x \tanh {\left (\frac {x}{2} \right )}}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} - \frac {2 b}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} - \frac {i x \sqrt {b^{2}}}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} & \text {for}\: a = - \sqrt {- b^{2}} \\\frac {b x \tanh {\left (\frac {x}{2} \right )}}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} - \frac {2 b}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} + \frac {i x \sqrt {b^{2}}}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} & \text {for}\: a = \sqrt {- b^{2}} \\\frac {a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} - \frac {a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} + \frac {x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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