Integrand size = 21, antiderivative size = 509 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {9 b d e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {5 b e^3 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {3 b d e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {5 b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e^3 x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {3 b d^2 e \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}-\frac {5 b e^3 \text {arccosh}(c x)}{96 c^6}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-3/4*b*d^2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-9/32*b*d*e^2*x*(c*x-1)^(1/2)* (c*x+1)^(1/2)/c^3-5/96*b*e^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-3/16*b*d*e^ 2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/144*b*e^3*x^3*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/c^3-1/36*b*e^3*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-3/4*b*d^2*e*arccosh (c*x)/c^2-9/32*b*d*e^2*arccosh(c*x)/c^4-5/96*b*e^3*arccosh(c*x)/c^6+3/2*d^ 2*e*x^2*(a+b*arccosh(c*x))+3/4*d*e^2*x^4*(a+b*arccosh(c*x))+1/6*e^3*x^6*(a +b*arccosh(c*x))-1/2*I*b*d^3*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2/(c*x-1)^(1/2 )/(c*x+1)^(1/2)+b*d^3*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2 +1)^(1/2))^2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+d^3*(a+b*arccosh(c*x))*ln(x)-b*d ^3*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*I* b*d^3*(-c^2*x^2+1)^(1/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/(c*x-1)^( 1/2)/(c*x+1)^(1/2)
Time = 0.70 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {3}{2} a d^2 e x^2+\frac {3}{4} a d e^2 x^4+\frac {1}{6} a e^3 x^6-\frac {3 b d^2 e \left (c x \sqrt {-1+c x} \sqrt {1+c x}-2 c^2 x^2 \text {arccosh}(c x)+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{4 c^2}-\frac {3 b d e^2 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )-8 c^4 x^4 \text {arccosh}(c x)+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^4}-\frac {b e^3 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (15+15 c x+10 c^2 x^2+10 c^3 x^3+8 c^4 x^4+8 c^5 x^5\right )-48 c^6 x^6 \text {arccosh}(c x)+30 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{288 c^6}+a d^3 \log (x)+\frac {1}{2} b d^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \] Input:
Integrate[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x,x]
Output:
(3*a*d^2*e*x^2)/2 + (3*a*d*e^2*x^4)/4 + (a*e^3*x^6)/6 - (3*b*d^2*e*(c*x*Sq rt[-1 + c*x]*Sqrt[1 + c*x] - 2*c^2*x^2*ArcCosh[c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(4*c^2) - (3*b*d*e^2*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*( 3 + 3*c*x + 2*c^2*x^2 + 2*c^3*x^3) - 8*c^4*x^4*ArcCosh[c*x] + 6*ArcTanh[Sq rt[(-1 + c*x)/(1 + c*x)]]))/(32*c^4) - (b*e^3*(c*x*Sqrt[(-1 + c*x)/(1 + c* x)]*(15 + 15*c*x + 10*c^2*x^2 + 10*c^3*x^3 + 8*c^4*x^4 + 8*c^5*x^5) - 48*c ^6*x^6*ArcCosh[c*x] + 30*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(288*c^6) + a*d^3*Log[x] + (b*d^3*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCo sh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2
Time = 1.51 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int \frac {2 e^3 x^6+9 d e^2 x^4+18 d^2 e x^2+12 d^3 \log (x)}{12 \sqrt {c x-1} \sqrt {c x+1}}dx+d^3 \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{12} b c \int \frac {2 e^3 x^6+9 d e^2 x^4+18 d^2 e x^2+12 d^3 \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}dx+d^3 \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{12} b c \int \left (\frac {2 e^3 x^6}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {9 d e^2 x^4}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {18 d^2 e x^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {12 d^3 \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx+d^3 \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {5 e^3 \text {arccosh}(c x)}{8 c^7}+\frac {27 d e^2 \text {arccosh}(c x)}{8 c^5}+\frac {9 d^2 e \text {arccosh}(c x)}{c^3}+\frac {6 i d^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {6 i d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {12 d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {12 d^3 \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 e^3 x \sqrt {c x-1} \sqrt {c x+1}}{8 c^6}+\frac {27 d e^2 x \sqrt {c x-1} \sqrt {c x+1}}{8 c^4}+\frac {5 e^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{12 c^4}+\frac {9 d^2 e x \sqrt {c x-1} \sqrt {c x+1}}{c^2}+\frac {9 d e^2 x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}+\frac {e^3 x^5 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\) |
Input:
Int[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x,x]
Output:
(3*d^2*e*x^2*(a + b*ArcCosh[c*x]))/2 + (3*d*e^2*x^4*(a + b*ArcCosh[c*x]))/ 4 + (e^3*x^6*(a + b*ArcCosh[c*x]))/6 + d^3*(a + b*ArcCosh[c*x])*Log[x] - ( b*c*((9*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^2 + (27*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(8*c^4) + (5*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(8*c^ 6) + (9*d*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c^2) + (5*e^3*x^3*Sqrt[ -1 + c*x]*Sqrt[1 + c*x])/(12*c^4) + (e^3*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) /(3*c^2) + (9*d^2*e*ArcCosh[c*x])/c^3 + (27*d*e^2*ArcCosh[c*x])/(8*c^5) + (5*e^3*ArcCosh[c*x])/(8*c^7) + ((6*I)*d^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2) /(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (12*d^3*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^3 *Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((6*I)*d^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/12
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 = 0.47 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.68
| method | result | size |
| parts | \(a \left (\frac {e^{3} x^{6}}{6}+\frac {3 d \,e^{2} x^{4}}{4}+\frac {3 d^{2} e \,x^{2}}{2}+d^{3} \ln \left (x \right )\right )+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}\) | \(348\) |
| derivativedivides | \(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {c^{6} e^{3} x^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}\) | \(365\) |
| default | \(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {c^{6} e^{3} x^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}\) | \(365\) |
Input:
int((e*x^2+d)^3*(a+b*arccosh(c*x))/x,x,method=_RETURNVERBOSE)
Output:
a*(1/6*e^3*x^6+3/4*d*e^2*x^4+3/2*d^2*e*x^2+d^3*ln(x))+1/2*b*d^3*polylog(2, -(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+d^3*b*arccosh(c*x)*ln(1+(c*x+(c*x-1) ^(1/2)*(c*x+1)^(1/2))^2)-3/4*b*d^2*e*arccosh(c*x)/c^2+1/6*b*arccosh(c*x)*e ^3*x^6-9/32*b*d*e^2*arccosh(c*x)/c^4-1/36*b*e^3*x^5*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/c-5/144*b*e^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-5/96*b*e^3*x*(c*x- 1)^(1/2)*(c*x+1)^(1/2)/c^5+3/4*b*arccosh(c*x)*d*e^2*x^4+3/2*b*arccosh(c*x) *d^2*e*x^2-3/16*b*d*e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-9/32*b*d*e^2*x*( c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/4*b*d^2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c -5/96*b*e^3*arccosh(c*x)/c^6-1/2*d^3*b*arccosh(c*x)^2
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x))/x,x, algorithm="fricas")
Output:
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, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x}\, dx \] Input:
integrate((e*x**2+d)**3*(a+b*acosh(c*x))/x,x)
Output:
Integral((a + b*acosh(c*x))*(d + e*x**2)**3/x, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x))/x,x, algorithm="maxima")
Output:
1/6*a*e^3*x^6 + 3/4*a*d*e^2*x^4 + 3/2*a*d^2*e*x^2 + a*d^3*log(x) + integra te(b*e^3*x^5*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*b*d*e^2*x^3*log(c* x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*b*d^2*e*x*log(c*x + sqrt(c*x + 1)*sqr t(c*x - 1)) + b*d^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x))/x,x, algorithm="giac")
Output:
integrate((e*x^2 + d)^3*(b*arccosh(c*x) + a)/x, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x} \,d x \] Input:
int(((a + b*acosh(c*x))*(d + e*x^2)^3)/x,x)
Output:
int(((a + b*acosh(c*x))*(d + e*x^2)^3)/x, x)
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {432 \mathit {acosh} \left (c x \right ) b \,c^{6} d^{2} e \,x^{2}+216 \mathit {acosh} \left (c x \right ) b \,c^{6} d \,e^{2} x^{4}+48 \mathit {acosh} \left (c x \right ) b \,c^{6} e^{3} x^{6}-216 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d^{2} e x -54 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d \,e^{2} x^{3}-8 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} e^{3} x^{5}-81 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} d \,e^{2} x -10 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} e^{3} x^{3}-15 \sqrt {c^{2} x^{2}-1}\, b c \,e^{3} x +288 \left (\int \frac {\mathit {acosh} \left (c x \right )}{x}d x \right ) b \,c^{6} d^{3}-216 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{4} d^{2} e -81 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{2} d \,e^{2}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,e^{3}+288 \,\mathrm {log}\left (x \right ) a \,c^{6} d^{3}+432 a \,c^{6} d^{2} e \,x^{2}+216 a \,c^{6} d \,e^{2} x^{4}+48 a \,c^{6} e^{3} x^{6}}{288 c^{6}} \] Input:
int((e*x^2+d)^3*(a+b*acosh(c*x))/x,x)
Output:
(432*acosh(c*x)*b*c**6*d**2*e*x**2 + 216*acosh(c*x)*b*c**6*d*e**2*x**4 + 4 8*acosh(c*x)*b*c**6*e**3*x**6 - 216*sqrt(c**2*x**2 - 1)*b*c**5*d**2*e*x - 54*sqrt(c**2*x**2 - 1)*b*c**5*d*e**2*x**3 - 8*sqrt(c**2*x**2 - 1)*b*c**5*e **3*x**5 - 81*sqrt(c**2*x**2 - 1)*b*c**3*d*e**2*x - 10*sqrt(c**2*x**2 - 1) *b*c**3*e**3*x**3 - 15*sqrt(c**2*x**2 - 1)*b*c*e**3*x + 288*int(acosh(c*x) /x,x)*b*c**6*d**3 - 216*log(sqrt(c**2*x**2 - 1) + c*x)*b*c**4*d**2*e - 81* log(sqrt(c**2*x**2 - 1) + c*x)*b*c**2*d*e**2 - 15*log(sqrt(c**2*x**2 - 1) + c*x)*b*e**3 + 288*log(x)*a*c**6*d**3 + 432*a*c**6*d**2*e*x**2 + 216*a*c* *6*d*e**2*x**4 + 48*a*c**6*e**3*x**6)/(288*c**6)