Optimal. Leaf size=432 \[ \frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2-b^2\right ) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d} \]
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Rubi [A]
time = 0.40, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5685, 3377,
2718, 5480, 3391, 30, 5681, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {2 x \left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 x \left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {x^2 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^3 d}+\frac {x^2 \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^3 d}-\frac {x^3 \left (a^2-b^2\right )}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{4 b d^3}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}+\frac {x^2}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 2718
Rule 3377
Rule 3391
Rule 5480
Rule 5681
Rule 5685
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac {a \int x^2 \sinh (c+d x) \, dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}+\frac {\left (a^2-b^2\right ) \int \frac {e^{c+d x} x^2}{a-\sqrt {a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {\left (a^2-b^2\right ) \int \frac {e^{c+d x} x^2}{a+\sqrt {a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(2 a) \int x \cosh (c+d x) \, dx}{b^2 d}-\frac {\int x \sinh ^2(c+d x) \, dx}{b d}\\ &=-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}-\frac {(2 a) \int \sinh (c+d x) \, dx}{b^2 d^2}+\frac {\int x \, dx}{2 b d}-\frac {\left (2 \left (a^2-b^2\right )\right ) \int x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}-\frac {\left (2 \left (a^2-b^2\right )\right ) \int x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}\\ &=\frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 \left (a^2-b^2\right )\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (2 \left (a^2-b^2\right )\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^2}\\ &=\frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}-\frac {\left (2 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}\\ &=\frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2-b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A]
time = 2.19, size = 697, normalized size = 1.61 \begin {gather*} \frac {\frac {8 \left (-a^2+b^2\right ) \left (2 d^3 e^{2 c} x^3-3 d^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-3 d^2 e^{2 c} x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-3 d^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-3 d^2 e^{2 c} x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-6 d \left (1+e^{2 c}\right ) x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-6 d \left (1+e^{2 c}\right ) x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )+6 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )+6 e^{2 c} \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )+6 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )+6 e^{2 c} \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )\right )}{d^3 \left (1+e^{2 c}\right )}-\frac {24 a b \cosh (d x) \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac {3 b^2 \cosh (2 d x) \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 b^2 \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^3}+8 \left (a^2-b^2\right ) x^3 \tanh (c)}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.79, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \cosh \left (d x +c \right )}\, dx\]
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. 1622 vs.
\(2 (402) = 804\).
time = 0.50, size = 1622, normalized size = 3.75 \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 {x^{2} \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\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 {x^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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