Optimal. Leaf size=327 \[ \frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^4}+\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^4}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b} \]
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Rubi [A] time = 0.48, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5562, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b d^2}-\frac {6 x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b d^3}+\frac {6 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^4}+\frac {6 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b d^4}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 5562
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac {x^4}{4 b}+\int \frac {e^{c+d x} x^3}{a-\sqrt {a^2-b^2}+b e^{c+d x}} \, dx+\int \frac {e^{c+d x} x^3}{a+\sqrt {a^2-b^2}+b e^{c+d x}} \, dx\\ &=-\frac {x^4}{4 b}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 \int x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d}-\frac {3 \int x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d}\\ &=-\frac {x^4}{4 b}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {6 \int x \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d^2}-\frac {6 \int x \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d^2}\\ &=-\frac {x^4}{4 b}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}+\frac {6 \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d^3}+\frac {6 \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d^3}\\ &=-\frac {x^4}{4 b}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b d^4}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b d^4}\\ &=-\frac {x^4}{4 b}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}+\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^4}+\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 326, normalized size = 1.00 \[ \frac {6 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}-a}\right )}{b d^4}+\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^4}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.57, size = 624, normalized size = 1.91 \[ -\frac {d^{4} x^{4} - 12 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - 12 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) + 4 \, c^{3} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) + 4 \, c^{3} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) + 24 \, d x {\rm polylog}\left (3, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) + 24 \, d x {\rm polylog}\left (3, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) - 4 \, {\left (d^{3} x^{3} + c^{3}\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - 4 \, {\left (d^{3} x^{3} + c^{3}\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - 24 \, {\rm polylog}\left (4, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) - 24 \, {\rm polylog}\left (4, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right )}{4 \, b d^{4}} \]
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 \[ \int \frac {x^{3} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sinh \left (d x +c \right )}{a +b \cosh \left (d x +c \right )}\, dx \]
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 \[ \frac {x^{4}}{4 \, b} - \frac {1}{2} \, \int \frac {4 \, {\left (a x^{3} e^{\left (d x + c\right )} + b x^{3}\right )}}{b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + b^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,\mathrm {sinh}\left (c+d\,x\right )}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {x^{3} \sinh {\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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