Optimal. Leaf size=51 \[ \frac {B \log (a+b \sinh (x))}{b}-\frac {2 A \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4401, 2660, 618, 206, 2668, 31} \[ \frac {B \log (a+b \sinh (x))}{b}-\frac {2 A \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 618
Rule 2660
Rule 2668
Rule 4401
Rubi steps
\begin {align*} \int \frac {A+B \cosh (x)}{a+b \sinh (x)} \, dx &=\int \left (\frac {A}{a+b \sinh (x)}+\frac {B \cosh (x)}{a+b \sinh (x)}\right ) \, dx\\ &=A \int \frac {1}{a+b \sinh (x)} \, dx+B \int \frac {\cosh (x)}{a+b \sinh (x)} \, dx\\ &=(2 A) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )+\frac {B \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac {B \log (a+b \sinh (x))}{b}-(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 )\\ &=-\frac {2 A \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {B \log (a+b \sinh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 59, normalized size = 1.16 \[ \frac {2 A \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {B \log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.17, size = 170, normalized size = 3.33 \[ \frac {\sqrt {a^{2} + b^{2}} A b \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) - {\left (B a^{2} + B b^{2}\right )} x + {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 87, normalized size = 1.71 \[ \frac {A \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} - \frac {B x}{b} + \frac {B \log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 1.73 \[ \frac {B \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{b}+\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) A}{\sqrt {a^{2}+b^{2}}}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 68, normalized size = 1.33 \[ \frac {A \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} + \frac {B \log \left (b \sinh \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 198, normalized size = 3.88 \[ \frac {B\,b^3\,\ln \left (8\,A^2\,a\,{\mathrm {e}}^x-4\,A^2\,b+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{a^2\,b^2+b^4}-\frac {B\,x}{b}-\frac {2\,\mathrm {atan}\left (\frac {A^2\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^2-b^2}}{\left (A\,a^2\,b+A\,b^3\right )\,\sqrt {A^2}}+\frac {A^2\,a\,b\,\sqrt {-a^2-b^2}}{\left (A\,a^2\,b+A\,b^3\right )\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {-a^2-b^2}}+\frac {B\,a^2\,b\,\ln \left (8\,A^2\,a\,{\mathrm {e}}^x-4\,A^2\,b+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{a^2\,b^2+b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 73.28, size = 745, normalized size = 14.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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