Optimal. Leaf size=527 \[ -\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3}+\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}+\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^4}-\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^4}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{6 f^3 \sinh (c+d x)}{b d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{b d} \]
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Rubi [A] time = 0.908079, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5565, 32, 3296, 2637, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3}+\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}+\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^4}-\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^4}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{6 f^3 \sinh (c+d x)}{b d^4}+\frac{(e+f x)^3 \cosh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 5565
Rule 32
Rule 3296
Rule 2637
Rule 3322
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac{a \int (e+f x)^3 \, dx}{b^2}+\frac{\int (e+f x)^3 \sinh (c+d x) \, dx}{b}+\frac{\left (a^2+b^2\right ) \int \frac{(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\left (2 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac{(3 f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b d}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac{\left (2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b}-\frac{\left (2 \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{b}+\frac{\left (6 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b d^2}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{\left (3 \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}+\frac{\left (3 \sqrt{a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}-\frac{\left (6 f^3\right ) \int \cosh (c+d x) \, dx}{b d^3}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{6 f^3 \sinh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac{\left (6 \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^2}+\frac{\left (6 \sqrt{a^2+b^2} f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^2}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^3}-\frac{6 f^3 \sinh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac{\left (6 \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^3}-\frac{\left (6 \sqrt{a^2+b^2} f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^3}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^3}-\frac{6 f^3 \sinh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac{\left (6 \sqrt{a^2+b^2} f^3\right ) \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^2 d^4}-\frac{\left (6 \sqrt{a^2+b^2} f^3\right ) \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^2 d^4}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac{(e+f x)^3 \cosh (c+d x)}{b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{3 \sqrt{a^2+b^2} f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2+b^2} f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{6 \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^4}-\frac{6 \sqrt{a^2+b^2} f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^4}-\frac{6 f^3 \sinh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \sinh (c+d x)}{b d^2}\\ \end{align*}
Mathematica [A] time = 3.28391, size = 933, normalized size = 1.77 \[ -\frac{a f^3 x^4 d^4+4 a e f^2 x^3 d^4+6 a e^2 f x^2 d^4+4 a e^3 x d^4+8 \sqrt{a^2+b^2} e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^3-4 b e^3 \cosh (c+d x) d^3-4 b f^3 x^3 \cosh (c+d x) d^3-12 b e f^2 x^2 \cosh (c+d x) d^3-12 b e^2 f x \cosh (c+d x) d^3-4 \sqrt{a^2+b^2} f^3 x^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-12 \sqrt{a^2+b^2} e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-12 \sqrt{a^2+b^2} e^2 f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+4 \sqrt{a^2+b^2} f^3 x^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+12 \sqrt{a^2+b^2} e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+12 \sqrt{a^2+b^2} e^2 f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-12 \sqrt{a^2+b^2} f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d^2+12 \sqrt{a^2+b^2} f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d^2+12 b f^3 x^2 \sinh (c+d x) d^2+12 b e^2 f \sinh (c+d x) d^2+24 b e f^2 x \sinh (c+d x) d^2-24 b e f^2 \cosh (c+d x) d-24 b f^3 x \cosh (c+d x) d+24 \sqrt{a^2+b^2} e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+24 \sqrt{a^2+b^2} f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d-24 \sqrt{a^2+b^2} e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d-24 \sqrt{a^2+b^2} f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d-24 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+24 \sqrt{a^2+b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )+24 b f^3 \sinh (c+d x)}{4 b^2 d^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.269, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.06394, size = 4691, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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