Optimal. Leaf size=325 \[ \frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d}-\frac {f^2 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^2}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d} \]
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Rubi [A] time = 0.65, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5569, 3716, 2190, 2531, 2282, 6589, 5561} \[ -\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a d^3}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^2}-\frac {f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5561
Rule 5569
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a}\\ &=-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d}\\ &=-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^2}-\frac {f^2 \int \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a d^2}\\ &=-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^2}-\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^3}+\frac {\left (2 f^2\right ) \operatorname {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 )}{a d^3}+\frac {\left (2 f^2\right ) \operatorname {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 )}{a d^3}\\ &=-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {f^2 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}\\ \end {align*}
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Mathematica [B] time = 5.75, size = 1013, normalized size = 3.12 \[ \frac {-\frac {2 (e+f x)^3}{\left (-1+e^{2 c}\right ) f}+\frac {3 \log \left (1-e^{-c-d x}\right ) (e+f x)^2}{d}+\frac {3 \log \left (1+e^{-c-d x}\right ) (e+f x)^2}{d}-\frac {6 f \left (d (e+f x) \text {Li}_2\left (-e^{-c-d x}\right )+f \text {Li}_3\left (-e^{-c-d x}\right )\right )}{d^3}-\frac {6 f \left (d (e+f x) \text {Li}_2\left (e^{-c-d x}\right )+f \text {Li}_3\left (e^{-c-d x}\right )\right )}{d^3}+\frac {2 e^{2 c} f^2 x^3 d^3+6 e e^{2 c} f x^2 d^3+6 e^2 e^{2 c} x d^3-3 e^2 e^{2 c} \log \left (-2 e^{c+d x} a-b e^{2 (c+d x)}+b\right ) d^2+3 e^2 \log \left (-2 e^{c+d x} a-b e^{2 (c+d x)}+b\right ) d^2-3 e^{2 c} f^2 x^2 \log \left (\frac {e^{2 c+d x} b}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2+3 f^2 x^2 \log \left (\frac {e^{2 c+d x} b}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-6 e e^{2 c} f x \log \left (\frac {e^{2 c+d x} b}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2+6 e f x \log \left (\frac {e^{2 c+d x} b}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-3 e^{2 c} f^2 x^2 \log \left (\frac {e^{2 c+d x} b}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2+3 f^2 x^2 \log \left (\frac {e^{2 c+d x} b}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-6 e e^{2 c} f x \log \left (\frac {e^{2 c+d x} b}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2+6 e f x \log \left (\frac {e^{2 c+d x} b}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-6 \left (-1+e^{2 c}\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d-6 \left (-1+e^{2 c}\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{2 c+d x}}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d+6 e^{2 c} f^2 \text {Li}_3\left (-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 f^2 \text {Li}_3\left (-\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} f^2 \text {Li}_3\left (-\frac {b e^{2 c+d x}}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 f^2 \text {Li}_3\left (-\frac {b e^{2 c+d x}}{e^c a+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3 \left (-1+e^{2 c}\right )}}{3 a} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.62, size = 813, normalized size = 2.50 \[ \frac {2 \, f^{2} {\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 ) + 2 \, f^{2} {\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 ) - 2 \, f^{2} {\rm polylog}\left (3, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - 2 \, f^{2} {\rm polylog}\left (3, -\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) - 2 \, {\left (d f^{2} x + d e f\right )} {\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 ) - 2 \, {\left (d f^{2} x + d e f\right )} {\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 ) + 2 \, {\left (d f^{2} x + d e f\right )} {\rm Li}_2\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left (d f^{2} x + d e f\right )} {\rm Li}_2\left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) - {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \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 ) - {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \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 ) - {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\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 ) - {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\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 ) + {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + d^{2} e^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\right )} \log \left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) + 1\right )}{a d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -e^{2} {\left (\frac {\log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} + \frac {2 \, {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {2 \, {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac {{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} - \frac {2 \, {\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2}\right )}}{3 \, a d^{3}} + \int -\frac {2 \, {\left (b f^{2} x^{2} + 2 \, b e f x - {\left (a f^{2} x^{2} e^{c} + 2 \, a e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} e^{\left (d x + c\right )} - a b}\,{d x} \]
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 \[ \int \frac {\mathrm {coth}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{2} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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