Integrand size = 21, antiderivative size = 476 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {3}{2} d e^2 x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {arccosh}(c x))-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{32 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+3 d^2 e (a+b \text {arccosh}(c x)) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/2*d^3*(a+b*arccosh(c*x))/x^2+3/2*d*e^2*x^2*(a+b*arccosh(c*x))+1/4*e^3*x ^4*(a+b*arccosh(c*x))+3*d^2*e*(a+b*arccosh(c*x))*ln(x)-1/2*b*c*d^3*(-c^2*x ^2+1)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/32*b*e^2*(8*c^2*d+e)*x*(-c^2*x^2+1)/ c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*e^3*x^3*(-c^2*x^2+1)/c/(c*x-1)^(1/2 )/(c*x+1)^(1/2)-3/2*I*b*d^2*e*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)+3*b*d^2*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)-3*b*d^2*e*arcsin(c*x)*ln(x) *(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/2*I*b*d^2*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) -3/32*b*e^2*(8*c^2*d+e)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c ^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.56 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.58 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a d^3}{x^2}+6 a d e^2 x^2+a e^3 x^4+\frac {2 b d^3 \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+6 b d e^2 x^2 \text {arccosh}(c x)+b e^3 x^4 \text {arccosh}(c x)-\frac {3 b d e^2 \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{c^2}-\frac {b e^3 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{8 c^4}+6 b d^2 e \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+12 a d^2 e \log (x)-6 b d^2 e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
((-2*a*d^3)/x^2 + 6*a*d*e^2*x^2 + a*e^3*x^4 + (2*b*d^3*(c*x*Sqrt[-1 + c*x] *Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2 + 6*b*d*e^2*x^2*ArcCosh[c*x] + b*e^3*x ^4*ArcCosh[c*x] - (3*b*d*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh [Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2 - (b*e^3*(c*x*Sqrt[(-1 + c*x)/(1 + c*x) ]*(3 + 3*c*x + 2*c^2*x^2 + 2*c^3*x^3) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x )]]))/(8*c^4) + 6*b*d^2*e*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*Arc Cosh[c*x])]) + 12*a*d^2*e*Log[x] - 6*b*d^2*e*PolyLog[2, -E^(-2*ArcCosh[c*x ])])/4
Time = 2.00 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int -\frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{4 x^2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^3 (a+b \text {arccosh}(c x))}{2 x^2}+3 d^2 e \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d e^2 x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} b c \int \frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^3 (a+b \text {arccosh}(c x))}{2 x^2}+3 d^2 e \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d e^2 x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} b c \int \left (\frac {-e^3 x^6-6 d e^2 x^4+2 d^3}{x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {12 d^2 e \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx-\frac {d^3 (a+b \text {arccosh}(c x))}{2 x^2}+3 d^2 e \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d e^2 x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \text {arccosh}(c x))}{2 x^2}+3 d^2 e \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d e^2 x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^3 x^4 (a+b \text {arccosh}(c x))+\frac {1}{4} b c \left (-\frac {6 i d^2 e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {6 i d^2 e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {12 d^2 e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {12 d^2 e \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 e^2 \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (8 c^2 d+e\right )}{8 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d^3 \left (1-c^2 x^2\right )}{x \sqrt {c x-1} \sqrt {c x+1}}+\frac {e^3 x^3 \left (1-c^2 x^2\right )}{4 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{8 c^4 \sqrt {c x-1} \sqrt {c x+1}}\right )\) |
-1/2*(d^3*(a + b*ArcCosh[c*x]))/x^2 + (3*d*e^2*x^2*(a + b*ArcCosh[c*x]))/2 + (e^3*x^4*(a + b*ArcCosh[c*x]))/4 + 3*d^2*e*(a + b*ArcCosh[c*x])*Log[x] + (b*c*((-2*d^3*(1 - c^2*x^2))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*e^2*( 8*c^2*d + e)*x*(1 - c^2*x^2))/(8*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (e^3* x^3*(1 - c^2*x^2))/(4*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((6*I)*d^2*e*Sqr t[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*e^2*(8 *c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(8*c^5*S qrt[-1 + c*x]*Sqrt[1 + c*x]) + (12*d^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]) - (12*d^2*e* Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ( (6*I)*d^2*e*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(c*Sqrt[- 1 + c*x]*Sqrt[1 + c*x])))/4
3.5.86.3.1 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 = 1.52 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.62
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{4}}{4}+\frac {3 d \,e^{2} x^{2}}{2}-\frac {d^{3}}{2 x^{2}}+3 d^{2} e \ln \left (x \right )\right )-\frac {b \,c^{2} d^{3}}{2}-\frac {3 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x d}{4 c}+\frac {b \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {3 b \,d^{2} e \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {3 b e \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right ) d}{4 c^{2}}+3 b e \,d^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b \,e^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3}}{16 c}-\frac {3 b \,e^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x}{32 c^{3}}+\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{2} d}{2}+\frac {b c \,d^{3} \sqrt {c x +1}\, \sqrt {c x -1}}{2 x}-\frac {3 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,d^{3} \operatorname {arccosh}\left (c x \right )}{2 x^{2}}\) | \(293\) |
| derivativedivides | \(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {c^{4} x^{4} e^{3}}{4}+3 c^{4} d^{2} e \ln \left (c x \right )-\frac {c^{4} d^{3}}{2 x^{2}}\right )}{c^{6}}-\frac {3 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{5}}-\frac {3 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c^{3}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{4}}{4 c^{2}}+\frac {b \,d^{3} \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}+\frac {3 b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d^{2} e}{2 c^{2}}-\frac {b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c^{3}}-\frac {d^{3} b}{2}-\frac {3 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}-\frac {3 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{32 c^{6}}-\frac {3 b \,d^{2} e \operatorname {arccosh}\left (c x \right )^{2}}{2 c^{2}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}+\frac {3 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d^{2} e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}\right )\) | \(331\) |
| default | \(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {c^{4} x^{4} e^{3}}{4}+3 c^{4} d^{2} e \ln \left (c x \right )-\frac {c^{4} d^{3}}{2 x^{2}}\right )}{c^{6}}-\frac {3 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{5}}-\frac {3 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c^{3}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{4}}{4 c^{2}}+\frac {b \,d^{3} \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}+\frac {3 b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d^{2} e}{2 c^{2}}-\frac {b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c^{3}}-\frac {d^{3} b}{2}-\frac {3 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}-\frac {3 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{32 c^{6}}-\frac {3 b \,d^{2} e \operatorname {arccosh}\left (c x \right )^{2}}{2 c^{2}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) d^{3}}{2 c^{2} x^{2}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}+\frac {3 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d^{2} e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}\right )\) | \(331\) |
a*(1/4*e^3*x^4+3/2*d*e^2*x^2-1/2*d^3/x^2+3*d^2*e*ln(x))-1/2*b*c^2*d^3-3/4* b/c*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*d+1/4*b*e^3*arccosh(c*x)*x^4-3/2*b*d ^2*e*arccosh(c*x)^2+3/2*b*e*d^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 ))^2)-3/4*b/c^2*e^2*arccosh(c*x)*d+3*b*e*d^2*arccosh(c*x)*ln(1+(c*x+(c*x-1 )^(1/2)*(c*x+1)^(1/2))^2)-1/16*b/c*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3-3/3 2*b/c^3*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+3/2*b*e^2*arccosh(c*x)*x^2*d+1/2 *b*c*d^3/x*(c*x+1)^(1/2)*(c*x-1)^(1/2)-3/32*b/c^4*e^3*arccosh(c*x)-1/2*b*d ^3/x^2*arccosh(c*x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d^3)*arccosh(c*x))/x^3, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/4*a*e^3*x^4 + 3/2*a*d*e^2*x^2 + 1/2*b*d^3*(sqrt(c^2*x^2 - 1)*c/x - arcco sh(c*x)/x^2) + 3*a*d^2*e*log(x) - 1/2*a*d^3/x^2 + integrate(b*e^3*x^3*log( c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*b*d*e^2*x*log(c*x + sqrt(c*x + 1)*s qrt(c*x - 1)) + 3*b*d^2*e*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^3} \,d x \]