Integrand size = 18, antiderivative size = 259 \[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^2 x+\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3}-\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c}-\frac {d^3 (a+b \text {arccosh}(c x))^2}{3 e}-\frac {d e (a+b \text {arccosh}(c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \text {arccosh}(c x))^2}{3 e} \]
2*b^2*d^2*x+4/9*b^2*e^2*x/c^2+1/2*b^2*d*e*x^2+2/27*b^2*e^2*x^3-1/3*d^3*(a+ b*arccosh(c*x))^2/e-1/2*d*e*(a+b*arccosh(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*ar ccosh(c*x))^2/e-2*b*d^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-4 /9*b*e^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-b*d*e*x*(a+b*a rccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-2/9*b*e^2*x^2*(a+b*arccosh(c*x) )*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.35 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.39 \[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=a^2 d^2 x+2 b^2 d^2 x+\frac {4 b^2 e^2 x}{9 c^2}+a^2 d e x^2+\frac {1}{2} b^2 d e x^2+\frac {1}{3} a^2 e^2 x^3+\frac {2}{27} b^2 e^2 x^3-\frac {2 a b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {4 a b e^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {a b d e x \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {2 a b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}-\frac {b \left (-6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \text {arccosh}(c x)}{9 c^3}+\frac {1}{6} b^2 \left (-\frac {3 d e}{c^2}+2 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text {arccosh}(c x)^2-\frac {a b d e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{c^2} \]
a^2*d^2*x + 2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + a^2*d*e*x^2 + (b^2*d*e*x ^2)/2 + (a^2*e^2*x^3)/3 + (2*b^2*e^2*x^3)/27 - (2*a*b*d^2*Sqrt[-1 + c*x]*S qrt[1 + c*x])/c - (4*a*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3) - (a*b* d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (2*a*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt [1 + c*x])/(9*c) - (b*(-6*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcCosh [c*x])/(9*c^3) + (b^2*((-3*d*e)/c^2 + 2*x*(3*d^2 + 3*d*e*x + e^2*x^2))*Arc Cosh[c*x]^2)/6 - (a*b*d*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/c^2
Time = 1.53 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6378, 6390, 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.
| \(\displaystyle \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arccosh}(c x))^2}{3 e}-\frac {2 b c \int \frac {(d+e x)^3 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 e}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arccosh}(c x))^2}{3 e}-\frac {2 b c \int \left (\frac {(a+b \text {arccosh}(c x)) d^3}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e x (a+b \text {arccosh}(c x)) d^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e^2 x^2 (a+b \text {arccosh}(c x)) d}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {e^3 x^3 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arccosh}(c x))^2}{3 e}-\frac {2 b c \left (\frac {2 e^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^4}+\frac {3 d e^2 (a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {3 d^2 e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {3 d e^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}+\frac {e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b c}-\frac {2 b e^3 x}{3 c^3}-\frac {3 b d^2 e x}{c}-\frac {3 b d e^2 x^2}{4 c}-\frac {b e^3 x^3}{9 c}\right )}{3 e}\) |
((d + e*x)^3*(a + b*ArcCosh[c*x])^2)/(3*e) - (2*b*c*((-3*b*d^2*e*x)/c - (2 *b*e^3*x)/(3*c^3) - (3*b*d*e^2*x^2)/(4*c) - (b*e^3*x^3)/(9*c) + (3*d^2*e*S qrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2 + (2*e^3*Sqrt[-1 + c *x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^4) + (3*d*e^2*x*Sqrt[-1 + c*x ]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^2) + (e^3*x^2*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^2) + (d^3*(a + b*ArcCosh[c*x])^2)/( 2*b*c) + (3*d*e^2*(a + b*ArcCosh[c*x])^2)/(4*b*c^3)))/(3*e)
3.1.22.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 0.84 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.66
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+\frac {c d e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{2}+c^{2} d^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arccosh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arccosh}\left (c x \right ) c^{3} d \,x^{2}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 c^{3} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+18 c^{2} d^{2} e \sqrt {c^{2} x^{2}-1}+9 c^{2} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+2 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 c d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+4 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{18 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c}\) | \(431\) |
| default | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+\frac {c d e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{2}+c^{2} d^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arccosh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arccosh}\left (c x \right ) c^{3} d \,x^{2}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 c^{3} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+18 c^{2} d^{2} e \sqrt {c^{2} x^{2}-1}+9 c^{2} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+2 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 c d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+4 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{18 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c}\) | \(431\) |
| parts | \(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d^{2} x +54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d e \,x^{2}+18 \operatorname {arccosh}\left (c x \right )^{2} e^{2} c^{3} x^{3}-108 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d e x -12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-27 \operatorname {arccosh}\left (c x \right )^{2} c d e -24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+108 c^{3} x \,d^{2}+27 c^{3} x^{2} d e +4 c^{3} x^{3} e^{2}+24 c x \,e^{2}\right )}{54 c^{3}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{3}}{3 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{2}+c \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}+\frac {c \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 c^{3} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+18 c^{2} d^{2} e \sqrt {c^{2} x^{2}-1}+9 c^{2} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+2 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 c d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+4 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} e \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(441\) |
1/c*(1/3*a^2/c^2*(c*e*x+c*d)^3/e+b^2/c^2*(1/27*e^2*(9*arccosh(c*x)^2*x^3*c ^3-6*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-12*arccosh(c*x)*(c*x -1)^(1/2)*(c*x+1)^(1/2)+2*c^3*x^3+12*c*x)+1/2*c*d*e*(2*arccosh(c*x)^2*x^2* c^2-2*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c*x-arccosh(c*x)^2+c^2*x^2) +c^2*d^2*(arccosh(c*x)^2*x*c-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2* c*x))+2*a*b/c^2*(1/3/e*arccosh(c*x)*c^3*d^3+arccosh(c*x)*c^3*d^2*x+e*arcco sh(c*x)*c^3*d*x^2+1/3*arccosh(c*x)*e^2*c^3*x^3-1/18/e*(c*x-1)^(1/2)*(c*x+1 )^(1/2)*(6*c^3*d^3*ln(c*x+(c^2*x^2-1)^(1/2))+18*c^2*d^2*e*(c^2*x^2-1)^(1/2 )+9*c^2*d*e^2*x*(c^2*x^2-1)^(1/2)+2*e^3*c^2*x^2*(c^2*x^2-1)^(1/2)+9*c*d*e^ 2*ln(c*x+(c^2*x^2-1)^(1/2))+4*e^3*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.23 \[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {2 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d e x^{2} + 9 \, {\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x - 3 \, b^{2} c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} + 4 \, b^{2} c e^{2}\right )} x + 6 \, {\left (6 \, a b c^{3} e^{2} x^{3} + 18 \, a b c^{3} d e x^{2} + 18 \, a b c^{3} d^{2} x - 9 \, a b c d e - {\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} + 4 \, b^{2} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} + 4 \, a b e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{54 \, c^{3}} \]
1/54*(2*(9*a^2 + 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 + b^2)*c^3*d*e*x^2 + 9*(2* b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2 + 6*b^2*c^3*d^2*x - 3*b^2*c*d*e)*log(c *x + sqrt(c^2*x^2 - 1))^2 + 6*(9*(a^2 + 2*b^2)*c^3*d^2 + 4*b^2*c*e^2)*x + 6*(6*a*b*c^3*e^2*x^3 + 18*a*b*c^3*d*e*x^2 + 18*a*b*c^3*d^2*x - 9*a*b*c*d*e - (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*d*e*x + 18*b^2*c^2*d^2 + 4*b^2*e^2)*sqrt (c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(2*a*b*c^2*e^2*x^2 + 9*a*b *c^2*d*e*x + 18*a*b*c^2*d^2 + 4*a*b*e^2)*sqrt(c^2*x^2 - 1))/c^3
\[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}\, dx \]
\[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
1/3*a^2*e^2*x^3 + b^2*d^2*x*arccosh(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arccosh( c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c ^3))*a*b*d*e + 2/9*(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))*a*b*e^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh (c*x)/c) + a^2*d^2*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^2/c + 1/3*(b^2*e^2*x^3 + 3*b^2*d*e*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) ^2 - integrate(2/3*(b^2*c^3*e^2*x^5 + 3*b^2*c^3*d*e*x^4 - b^2*c*e^2*x^3 - 3*b^2*c*d*e*x^2 + (b^2*c^2*e^2*x^4 + 3*b^2*c^2*d*e*x^3)*sqrt(c*x + 1)*sqrt (c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1) *sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)
Exception generated. \[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (d+e x)^2 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]