Optimal. Leaf size=55 \[ \frac {B x}{b}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 55, 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 = {2735, 2660, 618, 206} \[ \frac {B x}{b}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2735
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{a+b \sinh (x)} \, dx &=\frac {B x}{b}-\frac {(i (i A b-i a B)) \int \frac {1}{a+b \sinh (x)} \, dx}{b}\\ &=\frac {B x}{b}-\frac {(2 i (i A b-i a B)) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b}\\ &=\frac {B x}{b}+\frac {(4 i (i A b-i a B)) \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 {B x}{b}-\frac {2 (A b-a B) \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 = 0.11, size = 61, normalized size = 1.11 \[ \frac {\frac {2 (A b-a B) \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+B x}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 147, normalized size = 2.67 \[ -\frac {{\left (B a - A b\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) - {\left (B a^{2} + B b^{2}\right )} x}{a^{2} b + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 75, normalized size = 1.36 \[ \frac {B x}{b} - \frac {{\left (B a - A b\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 101, normalized size = 1.84 \[ \frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) A}{\sqrt {a^{2}+b^{2}}}-\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) a B}{b \sqrt {a^{2}+b^{2}}}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 124, normalized size = 2.25 \[ -B {\left (\frac {a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b} - \frac {x}{b}\right )} + \frac {A \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 269, normalized size = 4.89 \[ \frac {B\,x}{b}-\frac {2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^x\,\sqrt {-a^2\,b^2-b^4}\,\left (\frac {2\,\left (A\,b\,\sqrt {-a^2\,b^2-b^4}-B\,a\,\sqrt {-a^2\,b^2-b^4}\right )}{b^4\,\sqrt {-a^2\,b^2-b^4}\,\sqrt {{\left (A\,b-B\,a\right )}^2}}+\frac {2\,a^2\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{b^2\,\sqrt {-b^2\,\left (a^2+b^2\right )}\,\sqrt {-a^2\,b^2-b^4}\,\left (A\,b-B\,a\right )}\right )}{2}-\frac {a\,b\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {-b^2\,\left (a^2+b^2\right )}\,\left (A\,b-B\,a\right )}\right )\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {-a^2\,b^2-b^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 64.57, size = 422, normalized size = 7.67 \[ \begin {cases} \tilde {\infty } \left (A \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {A x + B \cosh {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {A \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x}{b} & \text {for}\: a = 0 \\- \frac {2 A b}{b^{2} + i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} + \frac {B b x}{b^{2} + i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} + \frac {i B x \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}}{b^{2} + i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} - \frac {2 i B \sqrt {b^{2}}}{b^{2} + i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = - \sqrt {- b^{2}} \\- \frac {2 A b}{b^{2} - i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} + \frac {B b x}{b^{2} - i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} - \frac {i B x \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}}{b^{2} - i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} + \frac {2 i B \sqrt {b^{2}}}{b^{2} - i b \sqrt {b^{2}} \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = \sqrt {- b^{2}} \\- \frac {A \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{\sqrt {a^{2} + b^{2}}} + \frac {A \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{\sqrt {a^{2} + b^{2}}} + \frac {B a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} - \frac {B a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} + \frac {B x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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