Optimal. Leaf size=53 \[ \frac {\sinh (c+d x)}{d (a+b)}+\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3676, 388, 205} \[ \frac {\sinh (c+d x)}{d (a+b)}+\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3676
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{(a+b) d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{(a+b) d}\\ &=\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{3/2} d}+\frac {\sinh (c+d x)}{(a+b) d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 53, normalized size = 1.00 \[ \frac {\sinh (c+d x)}{d (a+b)}+\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 766, normalized size = 14.45 \[ \left [\frac {{\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - \sqrt {-a^{2} - a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right ) - \cosh \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} + a + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) - a^{2} - a b}{2 \, {\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d \cosh \left (d x + c\right ) + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d \sinh \left (d x + c\right )\right )}}, \frac {{\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} + a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a^{2} + a b}}\right ) + 2 \, \sqrt {a^{2} + a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\frac {\sqrt {a^{2} + a b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, a}\right ) - a^{2} - a b}{2 \, {\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d \cosh \left (d x + c\right ) + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d \sinh \left (d x + c\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 1117, normalized size = 21.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 315, normalized size = 5.94 \[ -\frac {2}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \left (a +b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b^{2} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \left (a +b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {b^{2} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {2}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x\right )}}{2 \, {\left (a d e^{c} + b d e^{c}\right )}} + \frac {1}{2} \, \int \frac {4 \, {\left (b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}\right )}}{a^{2} + 2 \, a b + b^{2} + {\left (a^{2} e^{\left (4 \, c\right )} + 2 \, a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{2} e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.68, size = 154, normalized size = 2.91 \[ \frac {{\mathrm {e}}^{c+d\,x}}{2\,d\,\left (a+b\right )}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,d\,\left (a+b\right )}-\frac {b\,\ln \left (\sqrt {-a}\,\sqrt {a+b}+2\,a\,{\mathrm {e}}^{c+d\,x}-\sqrt {-a}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}\right )}{2\,\sqrt {-a}\,d\,{\left (a+b\right )}^{3/2}}+\frac {b\,\ln \left (2\,a\,{\mathrm {e}}^{c+d\,x}-\sqrt {-a}\,\sqrt {a+b}+\sqrt {-a}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}\right )}{2\,\sqrt {-a}\,d\,{\left (a+b\right )}^{3/2}} \]
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 \[ \int \frac {\cosh {\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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