Optimal. Leaf size=91 \[ \frac {a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b x}{6 c d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5716, 40, 39} \[ \frac {a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b x}{6 c d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 39
Rule 40
Rule 5716
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 c d^3}\\ &=\frac {b x}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 c d^3}\\ &=\frac {b x}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {b x}{6 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 64, normalized size = 0.70 \[ \frac {3 a+b c x \sqrt {c x-1} \sqrt {c x+1} \left (3-2 c^2 x^2\right )+3 b \cosh ^{-1}(c x)}{12 c^2 d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 98, normalized size = 1.08 \[ -\frac {3 \, a c^{4} x^{4} - 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} - d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 86, normalized size = 0.95 \[ \frac {\frac {a}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b \left (-\frac {\mathrm {arccosh}\left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {c x \left (2 c^{2} x^{2}-3\right )}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-1\right )}\right )}{d^{3}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, b {\left (\frac {4 \, \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 1}{c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} + 16 \, \int \frac {1}{4 \, {\left (c^{8} d^{3} x^{7} - 3 \, c^{6} d^{3} x^{5} + 3 \, c^{4} d^{3} x^{3} - c^{2} d^{3} x + {\left (c^{7} d^{3} x^{6} - 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} - c d^{3}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} + \frac {a}{4 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________