Integrand size = 21, antiderivative size = 839 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \text {arccosh}(c x)}{e^2}-\frac {d (a+b \text {arccosh}(c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \text {arccosh}(c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}} \] Output:
a*x/e^2-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e^2+b*x*arccosh(c*x)/e^2-1/4*d*(a+ b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)-e^(1/2)*x)+1/4*d*(a+b*arccosh(c*x))/e^ (5/2)/((-d)^(1/2)+e^(1/2)*x)+1/2*b*c*d*arctanh((c*(-d)^(1/2)-e^(1/2))^(1/2 )*(c*x+1)^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/(c*x-1)^(1/2))/(c*(-d)^(1/2)- e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/e^(5/2)-1/2*b*c*d*arctanh((c*( -d)^(1/2)+e^(1/2))^(1/2)*(c*x+1)^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1 )^(1/2))/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/e^(5/2) +3/4*(-d)^(1/2)*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^(5/2)-3/4*(-d)^(1/2)*(a+b*arcco sh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^ 2*d-e)^(1/2)))/e^(5/2)+3/4*(-d)^(1/2)*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x +(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e^(5/2)-3/4 *(-d)^(1/2)*(a+b*arccosh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e^(5/2)-3/4*b*(-d)^(1/2)*polylog(2,-e ^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/ e^(5/2)+3/4*b*(-d)^(1/2)*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 ))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^(5/2)-3/4*b*(-d)^(1/2)*polylog(2,-e^ (1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e ^(5/2)+3/4*b*(-d)^(1/2)*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) )/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e^(5/2)
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 777, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {8 a \sqrt {e} x+\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (\frac {8 \sqrt {e} \left (-\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+c x \text {arccosh}(c x)\right )}{c}+2 d \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+2 d \left (\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-3 i \sqrt {d} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+3 i \sqrt {d} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )}{8 e^{5/2}} \] Input:
Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]
Output:
(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt [e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(-(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + c*x*ArcCosh[c*x]))/c + 2*d*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c* Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]* Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^ 2*d) - e]) + 2*d*(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(-Sqr t[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) /(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - (3 *I)*Sqrt[d]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c *x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x] )/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[ c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^A rcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) + (3*I)*Sqrt[d]*(ArcCos h[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d ] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[ d] - Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqr t[d] + Sqrt[-(c^2*d) - e])])))/(8*e^(5/2))
Time = 2.61 (sec) , antiderivative size = 839, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6374, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )^2}-\frac {2 d (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \text {arccosh}(c x) b}{e^2}+\frac {c d \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {c d \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {3 \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {3 \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} b}{c e^2}+\frac {a x}{e^2}-\frac {d (a+b \text {arccosh}(c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \text {arccosh}(c x))}{4 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}}\) |
Input:
Int[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]
Output:
(a*x)/e^2 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) + (b*x*ArcCosh[c*x])/ e^2 - (d*(a + b*ArcCosh[c*x]))/(4*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b*ArcCosh[c*x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(Sq rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[- 1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/ 2)) - (b*c*d*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sq rt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*S qrt[-d] + Sqrt[e]]*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sq rt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*e^(5/2)) - (3 *Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d ] - Sqrt[-(c^2*d) - e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*L og[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*e^( 5/2)) - (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/ (c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(4*e^(5/2 )) + (3*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[ -(c^2*d) - e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCo sh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]* PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4 *e^(5/2))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 12.95 (sec) , antiderivative size = 897, normalized size of antiderivative = 1.07
| method | result | size |
| parts | \(a \left (\frac {x}{e^{2}}-\frac {d \left (-\frac {x}{2 \left (e \,x^{2}+d \right )}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{e^{2}}\right )+\frac {b \left (\frac {c^{4} \left (-1+\operatorname {arccosh}\left (c x \right )\right ) \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 e^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) c^{4} \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 e^{2}}+\frac {c^{7} \operatorname {arccosh}\left (c x \right ) d x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{6} d \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right ) c^{6} d}{2 e^{5}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{6} d \,\operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right ) c^{6} d}{2 e^{5}}+\frac {3 d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{4 e^{2}}-\frac {3 d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 e^{2}}\right )}{c^{5}}\) | \(897\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(913\) |
| default | \(\text {Expression too large to display}\) | \(913\) |
Input:
int(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
a*(1/e^2*x-1/e^2*d*(-1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2 ))))+b/c^5*(1/2*c^4*(-1+arccosh(c*x))/e^2*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) )+1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*c^4*(1+arccosh(c*x))/e^2+1/2*c^7* arccosh(c*x)/e^2*d*x/(c^2*e*x^2+c^2*d)+1/2*((2*c^2*d+2*(c^2*d*(c^2*d+e))^( 1/2)+e)*e)^(1/2)*(-2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(c^ 2*d*(c^2*d+e))^(1/2)*e)*c^6*d*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/( (2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^5/(c^2*d+e)-1/2*((2*c^2* d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2) +e)*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e ))^(1/2)+e)*e)^(1/2))*c^6*d/e^5+1/2*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e )*e)^(1/2)*(2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e+(c^2*d*(c^ 2*d+e))^(1/2)*e)*c^6*d*arctanh(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((-2*c^ 2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^5/(c^2*d+e)-1/2*(-(2*c^2*d-2* (c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)* arctanh(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e)) ^(1/2)-e)*e)^(1/2))*c^6*d/e^5+3/4*d/e^2*c^6*sum(1/_R1/(_R1^2*e+2*c^2*d+e)* (arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x -(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e ))-3/4*d/e^2*c^6*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c* x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/...
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*x^4*arccosh(c*x) + a*x^4)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**4*(a+b*acosh(c*x))/(e*x**2+d)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)*x^4/(e*x^2 + d)^2, x)
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^2,x)
Output:
int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^2, x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {-3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d -3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}+2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b d \,e^{3}+2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,e^{4} x^{2}+3 a d e x +2 a \,e^{2} x^{3}}{2 e^{3} \left (e \,x^{2}+d \right )} \] Input:
int(x^4*(a+b*acosh(c*x))/(e*x^2+d)^2,x)
Output:
( - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d - 3*sqrt(e)*sqrt(d )*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int((acosh(c*x)*x**4)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d*e**3 + 2*int((acosh(c*x)*x**4)/(d**2 + 2*d *e*x**2 + e**2*x**4),x)*b*e**4*x**2 + 3*a*d*e*x + 2*a*e**2*x**3)/(2*e**3*( d + e*x**2))