Optimal. Leaf size=212 \[ \frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b f \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {(e+f x) \sinh (c+d x)}{a d} \]
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Rubi [A]
time = 0.23, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5713, 5698,
3377, 2718, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {b f \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {b (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d}+\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}+\frac {(e+f x) \sinh (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 2718
Rule 3377
Rule 5680
Rule 5698
Rule 5713
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx &=\int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac {\int (e+f x) \cosh (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=\frac {b (e+f x)^2}{2 a^2 f}+\frac {(e+f x) \sinh (c+d x)}{a d}-\frac {b \int \frac {e^{c+d x} (e+f x)}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a}-\frac {f \int \sinh (c+d x) \, dx}{a d}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {(e+f x) \sinh (c+d x)}{a d}+\frac {(b f) \int \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {(b f) \int \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {(e+f x) \sinh (c+d x)}{a d}+\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}+\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b f \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {(e+f x) \sinh (c+d x)}{a d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.70, size = 435, normalized size = 2.05 \begin {gather*} \frac {\text {csch}(c+d x) (b+a \sinh (c+d x)) \left (-a f \cosh (c+d x)-b f (c+d x) \log (b+a \sinh (c+d x))+b c f \log \left (1+\frac {a \sinh (c+d x)}{b}\right )+i b f \left (-\frac {1}{8} i (2 c+i \pi +2 d x)^2-4 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(i a+b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {a^2+b^2}}\right )-\frac {1}{2} \left (-2 i c+\pi -2 i d x+4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )-\frac {1}{2} \left (-2 i c+\pi -2 i d x-4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )+\left (\frac {\pi }{2}-i (c+d x)\right ) \log (b+a \sinh (c+d x))+i \left (\text {PolyLog}\left (2,\frac {\left (b-\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )+\text {PolyLog}\left (2,\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )\right )\right )+a d f x \sinh (c+d x)+d e (-b \log (b+a \sinh (c+d x))+a \sinh (c+d x))\right )}{a^2 d^2 (a+b \text {csch}(c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs.
\(2(198)=396\).
time = 3.82, size = 483, normalized size = 2.28
method | result | size |
risch | \(\frac {b f \,x^{2}}{2 a^{2}}-\frac {b e x}{a^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 a \,d^{2}}-\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a \,d^{2}}+\frac {2 b e \ln \left ({\mathrm e}^{d x +c}\right )}{a^{2} d}-\frac {b e \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{a^{2} d}+\frac {2 b c f x}{a^{2} d}+\frac {b f \,c^{2}}{a^{2} d^{2}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}-\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}-\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}-\frac {b f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}-\frac {b f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}-\frac {2 b f c \ln \left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}+\frac {b f c \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{a^{2} d^{2}}\) | \(483\) |
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] Leaf count of result is larger than twice the leaf count of optimal. 780 vs.
\(2 (200) = 400\).
time = 0.48, size = 780, normalized size = 3.68 \begin {gather*} -\frac {a d f x + a d \cosh \left (1\right ) - {\left (a d f x + a d \cosh \left (1\right ) + a d \sinh \left (1\right ) - a f\right )} \cosh \left (d x + c\right )^{2} + a d \sinh \left (1\right ) - {\left (a d f x + a d \cosh \left (1\right ) + a d \sinh \left (1\right ) - a f\right )} \sinh \left (d x + c\right )^{2} + a f - {\left (b d^{2} f x^{2} - 2 \, b c^{2} f + 2 \, {\left (b d^{2} x + 2 \, b c d\right )} \cosh \left (1\right ) + 2 \, {\left (b d^{2} x + 2 \, b c d\right )} \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + 2 \, {\left (b f \cosh \left (d x + c\right ) + b f \sinh \left (d x + c\right )\right )} {\rm Li}_2\left (\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) + 2 \, {\left (b f \cosh \left (d x + c\right ) + b f \sinh \left (d x + c\right )\right )} {\rm Li}_2\left (\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) - {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) - 2 \, {\left ({\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (2 \, a \cosh \left (d x + c\right ) + 2 \, a \sinh \left (d x + c\right ) + 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, {\left ({\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (2 \, a \cosh \left (d x + c\right ) + 2 \, a \sinh \left (d x + c\right ) - 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, {\left ({\left (b d f x + b c f\right )} \cosh \left (d x + c\right ) + {\left (b d f x + b c f\right )} \sinh \left (d x + c\right )\right )} \log \left (-\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right ) + 2 \, {\left ({\left (b d f x + b c f\right )} \cosh \left (d x + c\right ) + {\left (b d f x + b c f\right )} \sinh \left (d x + c\right )\right )} \log \left (-\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) - {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right ) - {\left (b d^{2} f x^{2} - 2 \, b c^{2} f + 2 \, {\left (b d^{2} x + 2 \, b c d\right )} \cosh \left (1\right ) + 2 \, {\left (a d f x + a d \cosh \left (1\right ) + a d \sinh \left (1\right ) - a f\right )} \cosh \left (d x + c\right ) + 2 \, {\left (b d^{2} x + 2 \, b c d\right )} \sinh \left (1\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{2} d^{2} \cosh \left (d x + c\right ) + a^{2} d^{2} \sinh \left (d x + c\right )\right )}} \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 ) \cosh {\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\mathrm {cosh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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