Optimal. Leaf size=560 \[ \frac {2 a b^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a b^2 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.73, antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5692, 5680,
2221, 2317, 2438, 6874, 4265, 3799, 4270, 5559, 3852, 8} \begin {gather*} \frac {2 a b^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac {a (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {i a b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {i a b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac {b f \tanh (c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac {a f \text {sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {b^3 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )^2}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^2}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^2}-\frac {b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 3852
Rule 4265
Rule 4270
Rule 5559
Rule 5680
Rule 5692
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {b^2 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a^2+b^2}\\ &=-\frac {b^3 (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}+\frac {b^2 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int (e+f x) \text {sech}^3(c+d x) \, dx}{a^2+b^2}-\frac {b \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}\\ &=-\frac {b^3 (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (a b^2\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {b^3 \int (e+f x) \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int (e+f x) \text {sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}-\frac {\left (b^3 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (b^3 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {(b f) \int \text {sech}^2(c+d x) \, dx}{2 \left (a^2+b^2\right ) d}\\ &=\frac {2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {\left (2 b^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (b^3 f\right ) \text {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 )}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (b^3 f\right ) \text {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 )}{\left (a^2+b^2\right )^2 d^2}-\frac {(i b f) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 \left (a^2+b^2\right ) d^2}-\frac {\left (i a b^2 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (i a b^2 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {(i a f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 \left (a^2+b^2\right ) d}+\frac {(i a f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 \left (a^2+b^2\right ) d}\\ &=\frac {2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {\left (i a b^2 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (i a b^2 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {(i a f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {(i a f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {\left (b^3 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d}\\ &=\frac {2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {\left (b^3 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}\\ &=\frac {2 a b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \text {Li}_2\left (-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \text {Li}_2\left (i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 4.96, size = 588, normalized size = 1.05 \begin {gather*} \frac {2 b^3 d e (c+d x)-2 b^3 c f (c+d x)+2 a^3 d e \text {ArcTan}\left (e^{c+d x}\right )+6 a b^2 d e \text {ArcTan}\left (e^{c+d x}\right )-2 a^3 c f \text {ArcTan}\left (e^{c+d x}\right )-6 a b^2 c f \text {ArcTan}\left (e^{c+d x}\right )+i a^3 f (c+d x) \log \left (1-i e^{c+d x}\right )+3 i a b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )-i a^3 f (c+d x) \log \left (1+i e^{c+d x}\right )-3 i a b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b^3 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 b^3 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 b^3 d e \log \left (1+e^{2 (c+d x)}\right )+2 b^3 c f \log \left (1+e^{2 (c+d x)}\right )-2 b^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+2 b^3 d e \log (a+b \sinh (c+d x))-2 b^3 c f \log (a+b \sinh (c+d x))-i a \left (a^2+3 b^2\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )+i a \left (a^2+3 b^2\right ) f \text {PolyLog}\left (2,i e^{c+d x}\right )+2 b^3 f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 b^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-b^3 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+\left (a^2+b^2\right ) d (e+f x) \text {sech}^2(c+d x) (b+a \sinh (c+d x))+\left (a^2+b^2\right ) f \text {sech}(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2050 vs. \(2 (520 ) = 1040\).
time = 3.01, size = 2051, normalized size = 3.66
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2051\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 5468 vs. \(2 (512) = 1024\).
time = 0.44, size = 5468, normalized size = 9.76 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\left (e + f x\right ) \operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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