Optimal. Leaf size=278 \[ \frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d} \]
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Rubi [A]
time = 0.30, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5698, 5554,
2715, 8, 3377, 2718, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 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^3 d}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\frac {a f \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}+\frac {f x}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 2715
Rule 2718
Rule 3377
Rule 5554
Rule 5680
Rule 5698
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}-\frac {a \int (e+f x) \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int \sinh ^2(c+d x) \, dx}{2 b d}\\ &=-\frac {a^2 (e+f x)^2}{2 b^3 f}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 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}{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}{b^2}+\frac {(a f) \int \sinh (c+d x) \, dx}{b^2 d}+\frac {f \int 1 \, dx}{4 b d}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b 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^3 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^3 d}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}-\frac {\left (a^2 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 )}{b^3 d^2}-\frac {\left (a^2 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 )}{b^3 d^2}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 422, normalized size = 1.52 \begin {gather*} \frac {-2 b^2 d e \log (a+b \sinh (c+d x))+b^2 f \left (d x \left (d x-2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-2 \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+2 d e \left (b^2 \cosh (2 (c+d x))+\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b \sinh (c+d x)\right )+f \left (8 a b \cosh (c+d x)+2 b^2 d x \cosh (2 (c+d x))+2 \left (4 a^2+b^2\right ) \left (-\frac {1}{2} (c+d x)^2+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-c \log (a+b \sinh (c+d x))+\text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+\text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-8 a b d x \sinh (c+d x)-b^2 \sinh (2 (c+d x))\right )}{8 b^3 d^2} \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. \(564\) vs.
\(2(258)=516\).
time = 3.74, size = 565, normalized size = 2.03
method | result | size |
risch | \(-\frac {a^{2} f \,x^{2}}{2 b^{3}}+\frac {a^{2} e x}{b^{3}}+\frac {\left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 d^{2} b}-\frac {a \left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}+\frac {a \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}+\frac {\left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 d^{2} b}+\frac {2 a^{2} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{3}}-\frac {a^{2} f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{3}}-\frac {2 a^{2} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{3}}+\frac {a^{2} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{3}}-\frac {2 a^{2} c f x}{d \,b^{3}}-\frac {a^{2} f \,c^{2}}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}\) | \(565\) |
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. 1466 vs.
\(2 (261) = 522\).
time = 0.37, size = 1466, normalized size = 5.27 \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(-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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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 )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+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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