Integrand size = 19, antiderivative size = 251 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/2*d*(a+b*arccosh(c*x))/x^2+e*(a+b*arccosh(c*x))*ln(x)+1/2*b*c*d*(c*x-1) ^(1/2)*(c*x+1)^(1/2)/x-1/2*I*b*e*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)/(c*x-1)^ (1/2)/(c*x+1)^(1/2)+b*e*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c ^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*e*arcsin(c*x)*ln(x)*(-c^2*x^ 2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*I*b*e*polylog(2,(I*c*x+(-c^2*x^ 2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.40 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {a d}{2 x^2}+\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b d \text {arccosh}(c x)}{2 x^2}+a e \log (x)+\frac {1}{2} b e \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
-1/2*(a*d)/x^2 + (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (b*d*ArcCosh [c*x])/(2*x^2) + a*e*Log[x] + (b*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2
Time = 0.88 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6373, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int -\frac {\frac {d}{x^2}-2 e \log (x)}{2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e \log (x) (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b c \int \frac {\frac {d}{x^2}-2 e \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e \log (x) (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} b c \int \left (\frac {d}{x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 e \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e \log (x) (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e \log (x) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (-\frac {i e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {i e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 e \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \sqrt {c x-1} \sqrt {c x+1}}{x}\right )\) |
-1/2*(d*(a + b*ArcCosh[c*x]))/x^2 + e*(a + b*ArcCosh[c*x])*Log[x] + (b*c*( (d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x - (I*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2) /(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log [1 - E^((2*I)*ArcSin[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*e*Sqrt[ 1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*e*S qrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt [1 + c*x])))/2
3.5.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x _)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim p[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Time = 0.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\) | \(135\) |
| default | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\) | \(135\) |
| parts | \(-\frac {a d}{2 x^{2}}+a e \ln \left (x \right )+b \,c^{2} \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2 c^{2}}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\frac {e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{c^{2}}+\frac {e \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 c^{2}}\right )\) | \(135\) |
c^2*(a/c^2*e*ln(c*x)-1/2*a*d/c^2/x^2+b/c^2*(-1/2*e*arccosh(c*x)^2-1/2*d*(- (c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2+arccosh(c*x))/x^2+ln(1+(c*x+(c*x-1 )^(1/2)*(c*x+1)^(1/2))^2)*e*arccosh(c*x)+1/2*polylog(2,-(c*x+(c*x-1)^(1/2) *(c*x+1)^(1/2))^2)*e))
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/2*b*d*(sqrt(c^2*x^2 - 1)*c/x - arccosh(c*x)/x^2) + b*e*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x) + a*e*log(x) - 1/2*a*d/x^2
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^3} \,d x \]