Optimal. Leaf size=234 \[ \frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}+\frac {2 f^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^3}-\frac {2 f^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^3}-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))} \]
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Rubi [A]
time = 0.31, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5572, 3403,
2296, 2221, 2317, 2438} \begin {gather*} \frac {2 f^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^3 \sqrt {a^2+b^2}}-\frac {2 f^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^3 \sqrt {a^2+b^2}}+\frac {2 f (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d^2 \sqrt {a^2+b^2}}-\frac {2 f (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d^2 \sqrt {a^2+b^2}}-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 5572
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^2} \, dx &=-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}+\frac {(2 f) \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b d}\\ &=-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}+\frac {(4 f) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b d}\\ &=-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}+\frac {(4 f) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\sqrt {a^2+b^2} d}-\frac {(4 f) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\sqrt {a^2+b^2} d}\\ &=\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}-\frac {\left (2 f^2\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \sqrt {a^2+b^2} d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \sqrt {a^2+b^2} d^2}\\ &=\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \sqrt {a^2+b^2} d^3}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \sqrt {a^2+b^2} d^3}\\ &=\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {2 f (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}+\frac {2 f^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^3}-\frac {2 f^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^3}-\frac {(e+f x)^2}{b d (a+b \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 1.14, size = 175, normalized size = 0.75 \begin {gather*} \frac {2 f \left (d (e+f x) \left (\log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \sqrt {a^2+b^2} d^3}-\frac {(e+f x)^2}{b d (a+b \sinh (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. \(490\) vs.
\(2(214)=428\).
time = 6.45, size = 491, normalized size = 2.10
method | result | size |
risch | \(-\frac {2 \left (x^{2} f^{2}+2 e f x +e^{2}\right ) {\mathrm e}^{d x +c}}{b d \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}-\frac {4 f e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b \sqrt {a^{2}+b^{2}}}+\frac {2 f^{2} \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d^{2} b \sqrt {a^{2}+b^{2}}}+\frac {2 f^{2} \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{3} b \sqrt {a^{2}+b^{2}}}-\frac {2 f^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d^{2} b \sqrt {a^{2}+b^{2}}}-\frac {2 f^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{3} b \sqrt {a^{2}+b^{2}}}+\frac {2 f^{2} \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{3} b \sqrt {a^{2}+b^{2}}}-\frac {2 f^{2} \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{3} b \sqrt {a^{2}+b^{2}}}+\frac {4 f^{2} c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{3} b \sqrt {a^{2}+b^{2}}}\) | \(491\) |
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. 1590 vs.
\(2 (215) = 430\).
time = 0.38, size = 1590, normalized size = 6.79 \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 )\,{\left (e+f\,x\right )}^2}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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