Optimal. Leaf size=527 \[ \frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^3 \sinh (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac {(e+f x)^3 \cosh (c+d x)}{b d} \]
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Rubi [A] time = 0.91, antiderivative size = 527, normalized size of antiderivative = 1.00, 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 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3322
Rule 5565
Rule 6589
Rule 6609
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*}
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Mathematica [A] time = 3.09, size = 933, normalized size = 1.77 \[ -\frac {a f^3 x^4 d^4+4 a e f^2 x^3 d^4+6 a e^2 f x^2 d^4+4 a e^3 x d^4+8 \sqrt {a^2+b^2} e^3 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^3-4 b e^3 \cosh (c+d x) d^3-4 b f^3 x^3 \cosh (c+d x) d^3-12 b e f^2 x^2 \cosh (c+d x) d^3-12 b e^2 f x \cosh (c+d x) d^3-4 \sqrt {a^2+b^2} f^3 x^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-12 \sqrt {a^2+b^2} e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-12 \sqrt {a^2+b^2} e^2 f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+4 \sqrt {a^2+b^2} f^3 x^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+12 \sqrt {a^2+b^2} e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+12 \sqrt {a^2+b^2} e^2 f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-12 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d^2+12 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d^2+12 b f^3 x^2 \sinh (c+d x) d^2+12 b e^2 f \sinh (c+d x) d^2+24 b e f^2 x \sinh (c+d x) d^2-24 b e f^2 \cosh (c+d x) d-24 b f^3 x \cosh (c+d x) d+24 \sqrt {a^2+b^2} e f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+24 \sqrt {a^2+b^2} f^3 x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-24 \sqrt {a^2+b^2} e f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d-24 \sqrt {a^2+b^2} f^3 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d-24 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+24 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+24 b f^3 \sinh (c+d x)}{4 b^2 d^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.79, size = 2020, normalized size = 3.83 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cosh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -\frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )} a}{b^{2} d} - \frac {e^{\left (d x + c\right )}}{b d} - \frac {e^{\left (-d x - c\right )}}{b d} - \frac {2 \, \sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{2} d}\right )} - \frac {{\left (a d^{4} f^{3} x^{4} e^{c} + 4 \, a d^{4} e f^{2} x^{3} e^{c} + 6 \, a d^{4} e^{2} f x^{2} e^{c} - 2 \, {\left (b d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} b x^{2} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} b x e^{\left (2 \, c\right )} - 3 \, {\left (d^{2} e^{2} f - 2 \, d e f^{2} + 2 \, f^{3}\right )} b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - 2 \, {\left (b d^{3} f^{3} x^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} b x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} b x + 3 \, {\left (d^{2} e^{2} f + 2 \, d e f^{2} + 2 \, f^{3}\right )} b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{4 \, b^{2} d^{4}} + \int \frac {2 \, {\left ({\left (a^{2} f^{3} e^{c} + b^{2} f^{3} e^{c}\right )} x^{3} + 3 \, {\left (a^{2} e f^{2} e^{c} + b^{2} e f^{2} e^{c}\right )} x^{2} + 3 \, {\left (a^{2} e^{2} f e^{c} + b^{2} e^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}}{b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} e^{\left (d x + c\right )} - b^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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