Optimal. Leaf size=264 \[ d \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d \sqrt {1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1}}{4 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.68, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {14, 5790, 12, 6742, 90, 52, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {i b d \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+d \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d \sqrt {1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1}}{4 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 52
Rule 90
Rule 2190
Rule 2279
Rule 2326
Rule 2328
Rule 2391
Rule 3717
Rule 4625
Rule 5790
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {e x^2+2 d \log (x)}{2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {1}{2} (b c) \int \frac {e x^2+2 d \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {1}{2} (b c) \int \left (\frac {e x^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 d \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx\\ &=\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c d) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} (b c e) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {(b e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c}-\frac {\left (b c d \sqrt {1-c^2 x^2}\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b d \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \cosh ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {i b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 119, normalized size = 0.45 \[ \frac {1}{2} \left (2 a d \log (x)+a e x^2-\frac {b e \left (c x \sqrt {c x-1} \sqrt {c x+1}+2 \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )\right )}{2 c^2}-b d \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+b d \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+b e x^2 \cosh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcosh}\left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.32, size = 130, normalized size = 0.49 \[ \frac {a \,x^{2} e}{2}+d a \ln \left (c x \right )-\frac {b \mathrm {arccosh}\left (c x \right )^{2} d}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{2} e}{2}-\frac {b e \,\mathrm {arccosh}\left (c x \right )}{4 c^{2}}+b d \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a e x^{2} + a d \log \relax (x) + \int b e x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \frac {b d \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________