Integrand size = 27, antiderivative size = 278 \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/16*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/24*x^3*(a+b*arccosh( c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/6*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^( 1/2)+1/32*b*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/96* b*x^4*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/36*b*c*x^6*(-c^ 2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/32*(a+b*arccosh(c*x))^2*(-c ^2*d*x^2+d)^(1/2)/b/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.96 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.71 \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {48 a c x \sqrt {d-c^2 d x^2} \left (-3-2 c^2 x^2+8 c^4 x^4\right )-144 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{2304 c^5} \]
(48*a*c*x*Sqrt[d - c^2*d*x^2]*(-3 - 2*c^2*x^2 + 8*c^4*x^4) - 144*a*Sqrt[d] *ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCos h[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x] ] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/(2304*c^5)
Time = 1.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6341, 15, 6354, 15, 6354, 15, 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.
| \(\displaystyle \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^5dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/36*(b*c*x^6*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^5* Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/6 - (Sqrt[d - c^2*d*x^2]*(-1/16* (b*x^4)/c + (x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(4*c^2 ) + (3*(-1/4*(b*x^2)/c + (x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c* x]))/(2*c^2) + (a + b*ArcCosh[c*x])^2/(4*b*c^3)))/(4*c^2)))/(6*Sqrt[-1 + c *x]*Sqrt[1 + c*x])
3.1.58.3.1 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_.) + 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. \(877\) vs. \(2(234)=468\).
Time = 0.84 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.16
| method | result | size |
| default | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 c^{4} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{4}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(878\) |
| parts | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 c^{4} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{4}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(878\) |
-1/6*a*x^3*(-c^2*d*x^2+d)^(3/2)/c^2/d-1/8*a/c^4*x*(-c^2*d*x^2+d)^(3/2)/d+1 /16*a/c^4*x*(-c^2*d*x^2+d)^(1/2)+1/16*a/c^4*d/(c^2*d)^(1/2)*arctan((c^2*d) ^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-1/32*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)/c^5*arccosh(c*x)^2+1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*c^7* x^7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+38*c^3*x^3-48*(c*x+1 )^(1/2)*(c*x-1)^(1/2)*c^4*x^4-6*c*x+18*(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+6*arccosh(c*x))/(c*x+1)/c^5/(c*x-1)+1/51 2*(-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^5/(c*x-1)-1/256*(-d*(c^2*x^2-1))^( 1/2)*(2*c^3*x^3-2*c*x+2*(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+2*arccosh(c*x))/(c*x+1)/c^5/(c*x-1)-1/256*(-d*(c^2*x^2- 1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)* (c*x+1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))/(c*x+1)/c^5/(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) *(1+4*arccosh(c*x))/(c*x+1)/c^5/(c*x-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32 *(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+32*c^7*x^7+48*(c*x+1)^(1/2)*(c*x-1)^( 1/2)*c^4*x^4-64*c^5*x^5-18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+38*c^3*x^3+ (c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)*(1+6*arccosh(c*x))/(c*x+1)/c^5/(c*x-...
\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
-1/48*(8*(-c^2*d*x^2 + d)^(3/2)*x^3/(c^2*d) - 3*sqrt(-c^2*d*x^2 + d)*x/c^4 + 6*(-c^2*d*x^2 + d)^(3/2)*x/(c^4*d) - 3*sqrt(d)*arcsin(c*x)/c^5)*a + b*i ntegrate(sqrt(-c^2*d*x^2 + d)*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
Timed out. \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]