Optimal. Leaf size=516 \[ \frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}-\frac {a^2 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d \left (a^2+b^2\right )}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f} \]
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Rubi [A] time = 0.78, antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5581, 3718, 2190, 2279, 2391, 5567, 4180, 5573, 5561, 6742} \[ -\frac {i a^3 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {i a^3 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b d^2 \left (a^2+b^2\right )}-\frac {a^2 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2 \left (a^2+b^2\right )}+\frac {i a f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d \left (a^2+b^2\right )}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d \left (a^2+b^2\right )}-\frac {a^2 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d \left (a^2+b^2\right )}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d \left (a^2+b^2\right )}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 4180
Rule 5561
Rule 5567
Rule 5573
Rule 5581
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {a \int (e+f x) \text {sech}(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {a^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 d}-\frac {(i a f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b d}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {a^2 (e+f x)^2}{2 b \left (a^2+b^2\right ) f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2+b^2}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2+b^2}+\frac {a^2 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(i a f) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}-\frac {(i a f) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}-\frac {f \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b d^2}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {a^2 (e+f x)^2}{2 b \left (a^2+b^2\right ) f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {a^3 \int (e+f x) \text {sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {\left (2 a^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {\left (i a^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (i a^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {\left (i a^3 f\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (i a^3 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (a^2 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b \left (a^2+b^2\right ) d}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}+\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2}\\ &=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \text {Li}_2\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {a^2 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2}\\ \end {align*}
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Mathematica [A] time = 2.80, size = 438, normalized size = 0.85 \[ \frac {\frac {a^2 \left (f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac {1}{2} f (c+d x)^2\right )}{b}-2 a d e \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))+i a f \text {Li}_2(-i (\cosh (c+d x)+\sinh (c+d x)))-i a f \text {Li}_2(i (\cosh (c+d x)+\sinh (c+d x)))+2 a c f \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-2 a f (c+d x) \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-b d e (c+d x)+b d e \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)+\frac {1}{2} b f \text {Li}_2(-\cosh (2 (c+d x))-\sinh (2 (c+d x)))-\frac {1}{2} b f (c+d x)^2+b c f (c+d x)-b c f \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)+b f (c+d x) \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)}{d^2 \left (a^2+b^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 684, normalized size = 1.33 \[ -\frac {{\left (a^{2} + b^{2}\right )} d^{2} f x^{2} + 2 \, {\left (a^{2} + b^{2}\right )} d^{2} e x - 2 \, a^{2} f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 \, a^{2} f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, {\left (i \, a b f - b^{2} f\right )} {\rm Li}_2\left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right )\right ) + 2 \, {\left (-i \, a b f - b^{2} f\right )} {\rm Li}_2\left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right )\right ) - 2 \, {\left (a^{2} d e - a^{2} c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (a^{2} d e - a^{2} c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - {\left (-2 i \, a b d e + 2 \, b^{2} d e + 2 i \, a b c f - 2 \, b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + i\right ) - {\left (2 i \, a b d e + 2 \, b^{2} d e - 2 i \, a b c f - 2 \, b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - i\right ) - {\left (2 i \, a b d f x + 2 \, b^{2} d f x + 2 i \, a b c f + 2 \, b^{2} c f\right )} \log \left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right ) + 1\right ) - {\left (-2 i \, a b d f x + 2 \, b^{2} d f x - 2 i \, a b c f + 2 \, b^{2} c f\right )} \log \left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 3882, normalized size = 7.52 \[ \text {output too large to display} \]
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 \[ \frac {1}{2} \, f {\left (\frac {x^{2}}{b} - \int -\frac {4 \, {\left (a^{3} x e^{\left (d x + c\right )} - a^{2} b x\right )}}{a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} e^{\left (2 \, c\right )} + b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} b e^{c} + a b^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int \frac {4 \, {\left (a x e^{\left (d x + c\right )} + b x\right )}}{a^{2} + b^{2} + {\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x}\right )} + e {\left (\frac {a^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{2} b + b^{3}\right )} d} + \frac {2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {d x + c}{b d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {sinh}\left (c+d\,x\right )\,\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e + f x\right ) \sinh {\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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