Optimal. Leaf size=179 \[ -\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}-\frac {b \sqrt {c x-1}}{2 c^4 d^2 \sqrt {c x+1}}-\frac {b}{2 c^4 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \cosh ^{-1}(c x)}{2 c^4 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5750, 89, 12, 78, 52, 5715, 3716, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2}-\frac {b \sqrt {c x-1}}{2 c^4 d^2 \sqrt {c x+1}}-\frac {b}{2 c^4 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \cosh ^{-1}(c x)}{2 c^4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 52
Rule 78
Rule 89
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5715
Rule 5750
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac {\int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2 d}\\ &=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac {b \int \frac {c^2 x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^4 d^2}\\ &=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac {b \int \frac {x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^2 d^2}\\ &=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac {b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^3 d^2}\\ &=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \cosh ^{-1}(c x)}{2 c^4 d^2}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ &=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \cosh ^{-1}(c x)}{2 c^4 d^2}+\frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}+\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.68, size = 209, normalized size = 1.17 \[ \frac {-\frac {2 a}{c^2 x^2-1}+2 a \log \left (1-c^2 x^2\right )+4 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )+4 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )-b \sqrt {\frac {c x-1}{c x+1}}+\frac {b \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+\frac {b c x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}-2 b \cosh ^{-1}(c x)^2+\frac {b \cosh ^{-1}(c x)}{1-c x}+\frac {b \cosh ^{-1}(c x)}{c x+1}+4 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+4 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^4 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \operatorname {arcosh}\left (c x\right ) + a x^{3}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.61, size = 309, normalized size = 1.73 \[ -\frac {a}{4 c^{4} d^{2} \left (c x -1\right )}+\frac {a \ln \left (c x -1\right )}{2 c^{4} d^{2}}+\frac {a}{4 c^{4} d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{2 c^{4} d^{2}}-\frac {b \mathrm {arccosh}\left (c x \right )^{2}}{2 c^{4} d^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, x}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,x^{2}}{2 c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{4} d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{4} d^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{4} d^{2}}+\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, b {\left (\frac {{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 2 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )^{2} - 4 \, {\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right ) - 1\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 2}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - 8 \, \int \frac {{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right ) - 1}{2 \, {\left (c^{8} d^{2} x^{5} - 2 \, c^{6} d^{2} x^{3} + c^{4} d^{2} x + {\left (c^{7} d^{2} x^{4} - 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} - \frac {1}{2} \, a {\left (\frac {1}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________