Integrand size = 23, antiderivative size = 150 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{3 d} \]
4/9*b^2*e^2*x+2/27*b^2*e^2*(d*x+c)^3/d+1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+ c))^2/d-4/9*b*e^2*(a+b*arccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-2 /9*b*e^2*(d*x+c)^2*(a+b*arccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d
Time = 0.32 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.12 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e^2 \left (12 b^2 (c+d x)+\left (9 a^2+2 b^2\right ) (c+d x)^3+6 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2-(c+d x)^2\right )+6 b \left (3 a (c+d x)^3-2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+9 b^2 (c+d x)^3 \text {arccosh}(c+d x)^2\right )}{27 d} \]
(e^2*(12*b^2*(c + d*x) + (9*a^2 + 2*b^2)*(c + d*x)^3 + 6*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-2 - (c + d*x)^2) + 6*b*(3*a*(c + d*x)^3 - 2*b*Sq rt[-1 + c + d*x]*Sqrt[1 + c + d*x] - b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt [1 + c + d*x])*ArcCosh[c + d*x] + 9*b^2*(c + d*x)^3*ArcCosh[c + d*x]^2))/( 27*d)
Time = 0.66 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6411, 27, 6298, 6354, 15, 6330, 24}
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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{3} b \int (c+d x)^2d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b \int 1d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^2)/3 - (2*b*(-1/9*(b*(c + d*x) ^3) + (Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/3 + (2*(-(b*(c + d*x)) + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])))/3))/3))/d
3.2.6.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*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, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(167\) |
| default | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(167\) |
| parts | \(\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(172\) |
1/d*(1/3*a^2*e^2*(d*x+c)^3+e^2*b^2*(1/3*(d*x+c)^3*arccosh(d*x+c)^2-4/9*arc cosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2/9*(d*x+c)^2*arccosh(d*x+c)*( d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/9*d*x+4/9*c+2/27*(d*x+c)^3)+2*e^2*a*b*(1/ 3*(d*x+c)^3*arccosh(d*x+c)-1/9*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*((d*x+c)^2+ 2)))
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (132) = 264\).
Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.39 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} + 4 \, b^{2}\right )} d e^{2} x + 9 \, {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + b^{2} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 6 \, {\left (3 \, a b d^{3} e^{2} x^{3} + 9 \, a b c d^{2} e^{2} x^{2} + 9 \, a b c^{2} d e^{2} x + 3 \, a b c^{3} e^{2} - {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + {\left (b^{2} c^{2} + 2 \, b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 6 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + {\left (a b c^{2} + 2 \, a b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{27 \, d} \]
1/27*((9*a^2 + 2*b^2)*d^3*e^2*x^3 + 3*(9*a^2 + 2*b^2)*c*d^2*e^2*x^2 + 3*(( 9*a^2 + 2*b^2)*c^2 + 4*b^2)*d*e^2*x + 9*(b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^2 *x^2 + 3*b^2*c^2*d*e^2*x + b^2*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d *x + c^2 - 1))^2 + 6*(3*a*b*d^3*e^2*x^3 + 9*a*b*c*d^2*e^2*x^2 + 9*a*b*c^2* d*e^2*x + 3*a*b*c^3*e^2 - (b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + (b^2*c^2 + 2*b^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 6*(a*b*d^2*e^2*x^2 + 2*a*b*c*d*e^2*x + (a*b*c^2 + 2*a*b)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]
e**2*(Integral(a**2*c**2, x) + Integral(a**2*d**2*x**2, x) + Integral(b**2 *c**2*acosh(c + d*x)**2, x) + Integral(2*a*b*c**2*acosh(c + d*x), x) + Int egral(2*a**2*c*d*x, x) + Integral(b**2*d**2*x**2*acosh(c + d*x)**2, x) + I ntegral(2*a*b*d**2*x**2*acosh(c + d*x), x) + Integral(2*b**2*c*d*x*acosh(c + d*x)**2, x) + Integral(4*a*b*c*d*x*acosh(c + d*x), x))
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/3*a^2*d^2*e^2*x^3 + a^2*c*d*e^2*x^2 + (2*x^2*arccosh(d*x + c) - d*(3*c^2 *log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d ^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt (d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1) *c/d^3))*a*b*c*d*e^2 + 1/9*(6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2 *c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d) /d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c *d*x + c^2 - 1)*(c^2 - 1)/d^4))*a*b*d^2*e^2 + a^2*c^2*e^2*x + 2*((d*x + c) *arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a*b*c^2*e^2/d + 1/3*(b^2*d^2*e^ 2*x^3 + 3*b^2*c*d*e^2*x^2 + 3*b^2*c^2*e^2*x)*log(d*x + sqrt(d*x + c + 1)*s qrt(d*x + c - 1) + c)^2 - integrate(2/3*(b^2*d^5*e^2*x^5 + 5*b^2*c*d^4*e^2 *x^4 + (10*c^2*d^3*e^2 - d^3*e^2)*b^2*x^3 + 3*(3*c^3*d^2*e^2 - c*d^2*e^2)* b^2*x^2 + 3*(c^4*d*e^2 - c^2*d*e^2)*b^2*x + (b^2*d^4*e^2*x^4 + 4*b^2*c*d^3 *e^2*x^3 + 6*b^2*c^2*d^2*e^2*x^2 + 3*b^2*c^3*d*e^2*x)*sqrt(d*x + c + 1)*sq rt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(d^3*x ^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*s qrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \]