Integrand size = 18, antiderivative size = 241 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \sqrt {-1+c x} \sqrt {1+c x}}{35 c^7}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{105 c^7}-\frac {3 b e^2 \left (7 c^2 d+5 e\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c^7}-\frac {b e^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c^7}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x)) \] Output:
-1/35*b*(35*c^6*d^3+35*c^4*d^2*e+21*c^2*d*e^2+5*e^3)*(c*x-1)^(1/2)*(c*x+1) ^(1/2)/c^7-1/105*b*e*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(c*x-1)^(3/2)*(c*x+1)^ (3/2)/c^7-3/175*b*e^2*(7*c^2*d+5*e)*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^7-1/49*b *e^3*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c^7+d^3*x*(a+b*arccosh(c*x))+d^2*e*x^3*(a +b*arccosh(c*x))+3/5*d*e^2*x^5*(a+b*arccosh(c*x))+1/7*e^3*x^7*(a+b*arccosh (c*x))
Time = 0.16 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.80 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=a \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )}{3675 c^7}+\frac {1}{35} b x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \text {arccosh}(c x) \] Input:
Integrate[(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[-1 + c*x]* Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^ 4 + 75*e^3*x^6)))/(3675*c^7) + (b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x])/35
Time = 1.13 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6323, 27, 2113, 2331, 2389, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6323 |
| \(\displaystyle -b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{35} b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{\sqrt {c^2 x^2-1}}dx}{35 \sqrt {c x-1} \sqrt {c x+1}}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3}{\sqrt {c^2 x^2-1}}dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {5 \left (c^2 x^2-1\right )^{5/2} e^3}{c^6}+\frac {3 \left (7 d c^2+5 e\right ) \left (c^2 x^2-1\right )^{3/2} e^2}{c^6}+\frac {\left (35 d^2 c^4+42 d e c^2+15 e^2\right ) \sqrt {c^2 x^2-1} e}{c^6}+\frac {35 d^3 c^6+35 d^2 e c^4+21 d e^2 c^2+5 e^3}{c^6 \sqrt {c^2 x^2-1}}\right )dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 e^2 \left (c^2 x^2-1\right )^{5/2} \left (7 c^2 d+5 e\right )}{5 c^8}+\frac {10 e^3 \left (c^2 x^2-1\right )^{7/2}}{7 c^8}+\frac {2 e \left (c^2 x^2-1\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{3 c^8}+\frac {2 \sqrt {c^2 x^2-1} \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right )}{c^8}\right )}{70 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
-1/70*(b*c*Sqrt[-1 + c^2*x^2]*((2*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^ 2 + 5*e^3)*Sqrt[-1 + c^2*x^2])/c^8 + (2*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^ 2)*(-1 + c^2*x^2)^(3/2))/(3*c^8) + (6*e^2*(7*c^2*d + 5*e)*(-1 + c^2*x^2)^( 5/2))/(5*c^8) + (10*e^3*(-1 + c^2*x^2)^(7/2))/(7*c^8)))/(Sqrt[-1 + c*x]*Sq rt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) + d^2*e*x^3*(a + b*ArcCosh[c*x]) + (3*d*e^2*x^5*(a + b*ArcCosh[c*x]))/5 + (e^3*x^7*(a + b*ArcCosh[c*x]))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ m]/(a*c + b*d*x^2)^FracPart[m]) Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a *d, 0] && EqQ[m, n] && !IntegerQ[m]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(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}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.18 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.88
| method | result | size |
| parts | \(a \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{5}}{5}+c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 c^{6} d^{3}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675 c^{6}}\right )}{c}\) | \(211\) |
| derivativedivides | \(\frac {\frac {a \left (c^{7} d^{3} x +c^{7} d^{2} e \,x^{3}+\frac {3}{5} c^{7} d \,e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) c^{7} d^{3} x +\operatorname {arccosh}\left (c x \right ) c^{7} d^{2} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{7} d \,e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 c^{6} d^{3}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(235\) |
| default | \(\frac {\frac {a \left (c^{7} d^{3} x +c^{7} d^{2} e \,x^{3}+\frac {3}{5} c^{7} d \,e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) c^{7} d^{3} x +\operatorname {arccosh}\left (c x \right ) c^{7} d^{2} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{7} d \,e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 c^{6} d^{3}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(235\) |
| orering | \(\frac {x \left (325 e^{4} x^{8} c^{8}+1792 c^{8} d \,e^{3} x^{6}+4410 c^{8} d^{2} e^{2} x^{4}+30 c^{6} e^{4} x^{6}+9800 c^{8} d^{3} e \,x^{2}+294 c^{6} d \,e^{3} x^{4}+1225 c^{8} d^{4}+2450 c^{6} d^{2} e^{2} x^{2}+60 c^{4} e^{4} x^{4}-7350 c^{6} d^{3} e +1176 c^{4} d \,e^{3} x^{2}-4900 c^{4} d^{2} e^{2}+240 c^{2} e^{4} x^{2}-2352 c^{2} d \,e^{3}-480 e^{4}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{1225 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 c^{6} d^{3}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e x +\frac {\left (e \,x^{2}+d \right )^{3} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3675 c^{8} \left (e \,x^{2}+d \right )^{3}}\) | \(361\) |
Input:
int((e*x^2+d)^3*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/7*e^3*x^7+3/5*d*e^2*x^5+d^2*e*x^3+d^3*x)+b/c*(1/7*c*arccosh(c*x)*e^3* x^7+3/5*c*arccosh(c*x)*d*e^2*x^5+c*arccosh(c*x)*d^2*e*x^3+arccosh(c*x)*c*x *d^3-1/3675/c^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e^3*x^6+441*c^6*d*e^2* x^4+1225*c^6*d^2*e*x^2+90*c^4*e^3*x^4+3675*c^6*d^3+588*c^4*d*e^2*x^2+2450* c^4*d^2*e+120*c^2*e^3*x^2+1176*c^2*d*e^2+240*e^3))
Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \, {\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} + {\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^ 2*e*x^3 + 35*b*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*e^3*x^6 + 3675*b*c^6*d^3 + 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^ 2 + 10*b*c^4*e^3)*x^4 + 240*b*e^3 + (1225*b*c^6*d^2*e + 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^7
\[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \] Input:
integrate((e*x**2+d)**3*(a+b*acosh(c*x)),x)
Output:
Integral((a + b*acosh(c*x))*(d + e*x**2)**3, x)
Time = 0.03 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.19 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^2*e + 1/25*(1 5*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^ 2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d*e^2 + 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x ^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*e^3 + a*d^3*x + (c*x*arcc osh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c
Exception generated. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int((a + b*acosh(c*x))*(d + e*x^2)^3,x)
Output:
int((a + b*acosh(c*x))*(d + e*x^2)^3, x)
Time = 0.19 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.37 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {3675 \mathit {acosh} \left (c x \right ) b \,c^{7} d^{3} x +3675 \mathit {acosh} \left (c x \right ) b \,c^{7} d^{2} e \,x^{3}+2205 \mathit {acosh} \left (c x \right ) b \,c^{7} d \,e^{2} x^{5}+525 \mathit {acosh} \left (c x \right ) b \,c^{7} e^{3} x^{7}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} e \,x^{2}-441 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d \,e^{2} x^{4}-75 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{3} x^{6}-2450 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2} e -588 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,e^{2} x^{2}-90 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{3} x^{4}-1176 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d \,e^{2}-120 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{3} x^{2}-240 \sqrt {c^{2} x^{2}-1}\, b \,e^{3}-3675 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{6} d^{3}+3675 a \,c^{7} d^{3} x +3675 a \,c^{7} d^{2} e \,x^{3}+2205 a \,c^{7} d \,e^{2} x^{5}+525 a \,c^{7} e^{3} x^{7}}{3675 c^{7}} \] Input:
int((e*x^2+d)^3*(a+b*acosh(c*x)),x)
Output:
(3675*acosh(c*x)*b*c**7*d**3*x + 3675*acosh(c*x)*b*c**7*d**2*e*x**3 + 2205 *acosh(c*x)*b*c**7*d*e**2*x**5 + 525*acosh(c*x)*b*c**7*e**3*x**7 - 1225*sq rt(c**2*x**2 - 1)*b*c**6*d**2*e*x**2 - 441*sqrt(c**2*x**2 - 1)*b*c**6*d*e* *2*x**4 - 75*sqrt(c**2*x**2 - 1)*b*c**6*e**3*x**6 - 2450*sqrt(c**2*x**2 - 1)*b*c**4*d**2*e - 588*sqrt(c**2*x**2 - 1)*b*c**4*d*e**2*x**2 - 90*sqrt(c* *2*x**2 - 1)*b*c**4*e**3*x**4 - 1176*sqrt(c**2*x**2 - 1)*b*c**2*d*e**2 - 1 20*sqrt(c**2*x**2 - 1)*b*c**2*e**3*x**2 - 240*sqrt(c**2*x**2 - 1)*b*e**3 - 3675*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c**6*d**3 + 3675*a*c**7*d**3*x + 3675* a*c**7*d**2*e*x**3 + 2205*a*c**7*d*e**2*x**5 + 525*a*c**7*e**3*x**7)/(3675 *c**7)