Optimal. Leaf size=586 \[ \frac {3 x}{8 b d^3}+\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2-b^2\right ) x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2-b^2\right ) x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {6 \left (a^2-b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 a \sinh (c+d x)}{b^2 d^4}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d} \]
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Rubi [A]
time = 0.50, antiderivative size = 586, normalized size of antiderivative = 1.00, number of
steps used = 21, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used
= {5685, 3377, 2717, 5480, 3392, 30, 2715, 8, 5681, 2221, 2611, 6744, 2320, 6724}
\begin {gather*} \frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^4}-\frac {6 x \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 {6 x \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 {3 x^2 \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 {3 x^2 \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^3 \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^3 \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^4 \left (a^2-b^2\right )}{4 b^3}+\frac {6 a \sinh (c+d x)}{b^2 d^4}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {a x^3 \cosh (c+d x)}{b^2 d}-\frac {3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}-\frac {3 x^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 2715
Rule 2717
Rule 3377
Rule 3392
Rule 5480
Rule 5681
Rule 5685
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac {a \int x^3 \sinh (c+d x) \, dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (a^2-b^2\right ) \int \frac {e^{c+d x} x^3}{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^3}{a+\sqrt {a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(3 a) \int x^2 \cosh (c+d x) \, dx}{b^2 d}-\frac {3 \int x^2 \sinh ^2(c+d x) \, dx}{2 b d}\\ &=-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \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^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}-\frac {3 \int \sinh ^2(c+d x) \, dx}{4 b d^3}-\frac {(6 a) \int x \sinh (c+d x) \, dx}{b^2 d^2}+\frac {3 \int x^2 \, dx}{4 b d}-\frac {\left (3 \left (a^2-b^2\right )\right ) \int x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}-\frac {\left (3 \left (a^2-b^2\right )\right ) \int x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}\\ &=\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \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^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {(6 a) \int \cosh (c+d x) \, dx}{b^2 d^3}+\frac {3 \int 1 \, dx}{8 b d^3}-\frac {\left (6 \left (a^2-b^2\right )\right ) \int x \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (6 \left (a^2-b^2\right )\right ) \int x \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^2}\\ &=\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \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^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {6 a \sinh (c+d x)}{b^2 d^4}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 \left (a^2-b^2\right )\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^3}+\frac {\left (6 \left (a^2-b^2\right )\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d^3}\\ &=\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \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^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {6 a \sinh (c+d x)}{b^2 d^4}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 \left (a^2-b^2\right )\right ) \text {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^3 d^4}+\frac {\left (6 \left (a^2-b^2\right )\right ) \text {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^3 d^4}\\ &=\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \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^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2-b^2\right ) x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 a \sinh (c+d x)}{b^2 d^4}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A]
time = 2.69, size = 935, normalized size = 1.60 \begin {gather*} \frac {\frac {8 \left (-a^2+b^2\right ) \left (d^4 e^{2 c} x^4-2 d^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-2 d^3 e^{2 c} x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-2 d^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-2 d^3 e^{2 c} x^3 \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^2 \left (1+e^{2 c}\right ) x^2 \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^2 \left (1+e^{2 c}\right ) x^2 \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 )+12 d x \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 )+12 d e^{2 c} x \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 )+12 d x \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 )+12 d e^{2 c} x \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 )-12 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-12 e^{2 c} \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-12 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )-12 e^{2 c} \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2-b^2\right ) e^{2 c}}}\right )\right )}{d^4 \left (1+e^{2 c}\right )}-\frac {16 a b \cosh (d x) \left (d x \left (6+d^2 x^2\right ) \cosh (c)-3 \left (2+d^2 x^2\right ) \sinh (c)\right )}{d^4}+\frac {b^2 \cosh (2 d x) \left (2 d x \left (3+2 d^2 x^2\right ) \cosh (2 c)-3 \left (1+2 d^2 x^2\right ) \sinh (2 c)\right )}{d^4}-\frac {16 a b \left (-3 \left (2+d^2 x^2\right ) \cosh (c)+d x \left (6+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^4}+\frac {b^2 \left (-3 \left (1+2 d^2 x^2\right ) \cosh (2 c)+2 d x \left (3+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^4}+4 (a-b) (a+b) x^4 \tanh (c)}{16 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.81, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \cosh \left (d x +c \right )}\, dx\]
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. 2025 vs.
\(2 (546) = 1092\).
time = 0.48, size = 2025, normalized size = 3.46 \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^{3} \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^3\,{\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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