Optimal. Leaf size=60 \[ \frac {2 B \left (a^2-b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b \sqrt {a^2+b^2}}+\frac {B x}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2735, 2660, 618, 206} \[ \frac {2 B \left (a^2-b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b \sqrt {a^2+b^2}}+\frac {B x}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2735
Rubi steps
\begin {align*} \int \frac {\frac {b B}{a}+B \sinh (x)}{a+b \sinh (x)} \, dx &=\frac {B x}{b}-\frac {\left (i \left (-i a B+\frac {i b^2 B}{a}\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{b}\\ &=\frac {B x}{b}-\frac {\left (2 i \left (-i a B+\frac {i b^2 B}{a}\right )\right ) \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 {\left (4 i \left (-i a B+\frac {i b^2 B}{a}\right )\right ) \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 \left (a^2-b^2\right ) B \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b \sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 66, normalized size = 1.10 \[ \frac {B \left (a x-\frac {2 \left (a^2-b^2\right ) \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}\right )}{a b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 154, normalized size = 2.57 \[ -\frac {{\left (B a^{2} - B b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) - {\left (B a^{3} + B a b^{2}\right )} x}{a^{3} b + a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 82, normalized size = 1.37 \[ \frac {B x}{b} - \frac {{\left (B a^{2} - B b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 1.75 \[ -\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) a B}{b \sqrt {a^{2}+b^{2}}}+\frac {2 B b \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 128, normalized size = 2.13 \[ -B {\left (\frac {a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b} - \frac {x}{b}\right )} + \frac {B b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 331, normalized size = 5.52 \[ \frac {2\,\mathrm {atan}\left (\frac {a\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^4\,b^2-a^2\,b^4}\,\left (\frac {2\,\left (B\,a^2\,\sqrt {-a^4\,b^2-a^2\,b^4}-B\,b^2\,\sqrt {-a^4\,b^2-a^2\,b^4}\right )}{a^2\,b^4\,\sqrt {-a^4\,b^2-a^2\,b^4}\,\sqrt {B^2\,{\left (a^2-b^2\right )}^2}}+\frac {2\,a^2\,\sqrt {B^2\,a^4-2\,B^2\,a^2\,b^2+B^2\,b^4}}{B\,b^2\,\sqrt {-a^4\,b^2-a^2\,b^4}\,\left (a^2-b^2\right )\,\sqrt {-a^2\,b^2\,\left (a^2+b^2\right )}}\right )}{2}-\frac {a^2\,b\,\sqrt {B^2\,a^4-2\,B^2\,a^2\,b^2+B^2\,b^4}}{B\,\left (a^2-b^2\right )\,\sqrt {-a^2\,b^2\,\left (a^2+b^2\right )}}\right )\,\sqrt {B^2\,a^4-2\,B^2\,a^2\,b^2+B^2\,b^4}}{\sqrt {-a^4\,b^2-a^2\,b^4}}+\frac {B\,x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.55, size = 340, normalized size = 5.67 \[ \begin {cases} \text {NaN} & \text {for}\: a = 0 \wedge b = 0 \\\frac {B \cosh {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {B b x \tanh {\left (\frac {x}{2} \right )}}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} - \frac {4 B b}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} - \frac {i B x \sqrt {b^{2}}}{b^{2} \tanh {\left (\frac {x}{2} \right )} - i b \sqrt {b^{2}}} & \text {for}\: a = - \sqrt {- b^{2}} \\\frac {B b x \tanh {\left (\frac {x}{2} \right )}}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} - \frac {4 B b}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} + \frac {i B x \sqrt {b^{2}}}{b^{2} \tanh {\left (\frac {x}{2} \right )} + i b \sqrt {b^{2}}} & \text {for}\: a = \sqrt {- b^{2}} \\\frac {B a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} - \frac {B a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} + \frac {B x}{b} - \frac {B b \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a \sqrt {a^{2} + b^{2}}} + \frac {B b \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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