Integrand size = 29, antiderivative size = 319 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/64*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*b^2*x^3*(-c^2*d*x^2+d)^(1/2)-1/8 *x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+1/4*x^3*(a+b*arccosh(c*x) )^2*(-c^2*d*x^2+d)^(1/2)-1/64*b^2*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(c *x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2 )/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/8*b*c*x^4*(a+b*arccosh(c*x))*(-c^2*d*x^2 +d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/24*(a+b*arccosh(c*x))^3*(-c^2*d*x^ 2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 2.13 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.76 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {-96 a^2 c x \left (-1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}+96 a^2 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {12 a b \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{768 c^3} \]
-1/768*(-96*a^2*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] + 96*a^2*Sqrt[d]* ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (12*a*b*Sqrt[ d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*S inh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b^2*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x) ]*(1 + c*x)))/c^3
Time = 1.87 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {6341, 6298, 111, 27, 101, 43, 6354, 6298, 101, 43, 6308}
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.
| \(\displaystyle \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int x (a+b \text {arccosh}(c x))dx}{c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x))^3}{6 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\) |
(x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (b*c*Sqrt[d - c^2*d*x ^2]*((x^4*(a + b*ArcCosh[c*x]))/4 - (b*c*((x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x ])/(4*c^2) + (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/( 2*c^3)))/(4*c^2)))/4))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2*d* x^2]*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*c^2) + (a + b*ArcCosh[c*x])^3/(6*b*c^3) - (b*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*( (x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2))/c))/ (4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(677\) vs. \(2(271)=542\).
Time = 0.65 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \operatorname {arccosh}\left (c x \right )^{2}-4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (8 \operatorname {arccosh}\left (c x \right )^{2}+4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(678\) |
| parts | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \operatorname {arccosh}\left (c x \right )^{2}-4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (8 \operatorname {arccosh}\left (c x \right )^{2}+4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(678\) |
-1/4*a^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+1 /8*a^2/c^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^ 2*(-1/24*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c* x)^3+1/512*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c *x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1 /2)*(c*x+1)^(1/2))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)/(c*x+1)/c^3/(c*x-1) +1/512*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^ 5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+ 1)^(1/2)+4*c*x)*(8*arccosh(c*x)^2+4*arccosh(c*x)+1)/(c*x+1)/c^3/(c*x-1))+2 *a*b*(-1/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh (c*x)^2+1/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2) *(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1) ^(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c ^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c* x)*(1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1))
\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
integral((b^2*x^2*arccosh(c*x)^2 + 2*a*b*x^2*arccosh(c*x) + a^2*x^2)*sqrt( -c^2*d*x^2 + d), x)
\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/8*a^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d)*arcsin(c*x)/c^3) + integrate(sqrt(-c^2*d*x^2 + d)*b^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*sqrt(-c^2*d*x^2 + d)*a*b*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]