Optimal. Leaf size=69 \[ -\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \tanh (x)}{a^2+b^2}-\frac {b \text {sech}(x)}{a^2+b^2} \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2727, 3767, 8, 2606, 2660, 618, 206} \[ -\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \tanh (x)}{a^2+b^2}-\frac {b \text {sech}(x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 618
Rule 2606
Rule 2660
Rule 2727
Rule 3767
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx &=-\frac {a \int \text {sech}^2(x) \, dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a+b \sinh (x)} \, dx}{a^2+b^2}+\frac {b \int \text {sech}(x) \tanh (x) \, dx}{a^2+b^2}\\ &=-\frac {(i a) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))}{a^2+b^2}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}-\frac {b \operatorname {Subst}(\int 1 \, dx,x,\text {sech}(x))}{a^2+b^2}\\ &=-\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2}-\frac {\left (4 a^2\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 )}{a^2+b^2}\\ &=-\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 69, normalized size = 1.00 \[ \frac {a \left (\frac {2 a \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\tanh (x)\right )-b \text {sech}(x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 257, normalized size = 3.72 \[ \frac {2 \, a^{3} + 2 \, a b^{2} + {\left (a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) \sinh \relax (x) + a^{2} \sinh \relax (x)^{2} + a^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \relax (x) - 2 \, {\left (a^{2} b + b^{3}\right )} \sinh \relax (x)}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 1.26 \[ \frac {a^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{x} - a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 84, normalized size = 1.22 \[ \frac {8 a^{2} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 a \tanh \left (\frac {x}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 89, normalized size = 1.29 \[ \frac {a^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{\left (-x\right )} + a\right )}}{a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 330, normalized size = 4.78 \[ \frac {\frac {2\,a}{a^2+b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {2\,\mathrm {atan}\left (\left (\frac {b^3\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}{2}+\frac {a^2\,b\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}{2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^2\,\sqrt {a^4}\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (a^3\,\sqrt {a^4}+a\,b^2\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}\right )-\frac {2\,\left (b^3\,\sqrt {a^4}+a^2\,b\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}\right )\right )\,\sqrt {a^4}}{\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}} \]
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 \[ \int \frac {\tanh ^{2}{\relax (x )}}{a + b \sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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