Optimal. Leaf size=325 \[ -\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d}-\frac {(e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.51, 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 = {5688, 3797,
2221, 2611, 2320, 6724, 5680} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5680
Rule 5688
Rule 6724
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 \text {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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 4.25, size = 492, normalized size = 1.51 \begin {gather*} \frac {2 d^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )+2 d f (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )-f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )-2 \left (d^2 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \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^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \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 f (e+f x) \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 )+2 d f (e+f x) \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 )-2 f^2 \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 )-2 f^2 \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 )\right )}{2 a d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 990 vs.
\(2 (308) = 616\).
time = 0.40, size = 990, normalized size = 3.05 \begin {gather*} \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 f \cosh \left (1\right ) + d f \sinh \left (1\right )\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 f \cosh \left (1\right ) + d f \sinh \left (1\right )\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 f \cosh \left (1\right ) + d f \sinh \left (1\right )\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 f \cosh \left (1\right ) + d f \sinh \left (1\right )\right )} {\rm Li}_2\left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) - {\left (c^{2} f^{2} - 2 \, c d f \cosh \left (1\right ) + d^{2} \cosh \left (1\right )^{2} + d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (c d f - d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\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 (c^{2} f^{2} - 2 \, c d f \cosh \left (1\right ) + d^{2} \cosh \left (1\right )^{2} + d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (c d f - d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\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} - c^{2} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cosh \left (1\right ) + 2 \, {\left (d^{2} f x + c d f\right )} \sinh \left (1\right )\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} - c^{2} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cosh \left (1\right ) + 2 \, {\left (d^{2} f x + c d f\right )} \sinh \left (1\right )\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} f x \cosh \left (1\right ) + d^{2} \cosh \left (1\right )^{2} + d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (d^{2} f x + d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left (c^{2} f^{2} - 2 \, c d f \cosh \left (1\right ) + d^{2} \cosh \left (1\right )^{2} + d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (c d f - d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + {\left (d^{2} f^{2} x^{2} - c^{2} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cosh \left (1\right ) + 2 \, {\left (d^{2} f x + c d f\right )} \sinh \left (1\right )\right )} \log \left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) + 1\right )}{a d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{2} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________