Optimal. Leaf size=75 \[ -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+b c d \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {b e \sqrt {c x-1} \sqrt {c x+1}}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5786, 460, 92, 205} \[ -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+b c d \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {b e \sqrt {c x-1} \sqrt {c x+1}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 92
Rule 205
Rule 460
Rule 5786
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+(b c) \int \frac {d-e x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+(b c d) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+\left (b c^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+b c d \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 105, normalized size = 1.40 \[ -\frac {a d}{x}+a e x+\frac {b c d \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d \cosh ^{-1}(c x)}{x}-\frac {b e \sqrt {c x-1} \sqrt {c x+1}}{c}+b e x \cosh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 132, normalized size = 1.76 \[ \frac {2 \, b c^{2} d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a c e x^{2} - \sqrt {c^{2} x^{2} - 1} b e x - a c d + {\left (b c d - b c e\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 95, normalized size = 1.27 \[ a x e -\frac {a d}{x}+b \,\mathrm {arccosh}\left (c x \right ) x e -\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{x}-\frac {c b \sqrt {c x -1}\, \sqrt {c x +1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {b e \sqrt {c x -1}\, \sqrt {c x +1}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 63, normalized size = 0.84 \[ -{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e}{c} - \frac {a d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________