Integrand size = 27, antiderivative size = 360 \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/8*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-3/128*d*x*(a+b*arccosh(c*x ))*(-c^2*d*x^2+d)^(1/2)/c^4-1/64*d*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^( 1/2)/c^2+1/16*d*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/256*b*d*x^2* (-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/256*b*d*x^4*(-c^2*d *x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/32*b*c*d*x^6*(-c^2*d*x^2+d)^ (1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/64*b*c^3*d*x^8*(-c^2*d*x^2+d)^(1/2)/(c *x-1)^(1/2)/(c*x+1)^(1/2)-3/256*d*(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 = 3.47 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.94 \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-576 a c x \sqrt {d-c^2 d x^2} \left (3+2 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )-1728 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {32 b \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b \sqrt {d-c^2 d x^2} \left (1440 \text {arccosh}(c x)^2-576 \cosh (2 \text {arccosh}(c x))+144 \cosh (4 \text {arccosh}(c x))+64 \cosh (6 \text {arccosh}(c x))+9 \cosh (8 \text {arccosh}(c x))-24 \text {arccosh}(c x) (-48 \sinh (2 \text {arccosh}(c x))+24 \sinh (4 \text {arccosh}(c x))+16 \sinh (6 \text {arccosh}(c x))+3 \sinh (8 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{73728 c^5} \]
(d*(-576*a*c*x*Sqrt[d - c^2*d*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^ 6) - 1728*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^ 2))] + (32*b*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c *x]] - 9*Cosh[4*ArcCosh[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)) + (b*Sqrt[d - c^2*d*x^2]*(1440*Arc Cosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cos h[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] - 24*ArcCosh[c*x]*(-48*Sinh[2*A rcCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[ 8*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(73728*c^5)
Time = 1.51 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6345, 25, 82, 244, 2009, 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.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x^5 (1-c x) (c x+1)dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^5 (1-c x) (c x+1)dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^5 \left (1-c^2 x^2\right )dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^5-c^2 x^7\right )dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^5dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {3}{8} d \left (\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/8*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^6/6 - (c^2*x^8)/8))/(Sqrt[-1 + c*x]*Sqr t[1 + c*x]) + (x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/8 + (3*d*(- 1/36*(b*c*x^6*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^5*S qrt[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])))/8
3.1.71.3.1 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_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -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. \(782\) vs. \(2(304)=608\).
Time = 0.75 (sec) , antiderivative size = 783, normalized size of antiderivative = 2.18
| method | result | size |
| default | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 c^{9} x^{9}-320 c^{7} x^{7}+128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+272 c^{5} x^{5}-256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-88 c^{3} x^{3}+160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c x -32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \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 ) d}{1024 \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 ) d}{1024 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+128 c^{9} x^{9}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+272 c^{5} x^{5}+32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-88 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+8 c x \right ) \left (1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(783\) |
| parts | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 c^{9} x^{9}-320 c^{7} x^{7}+128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+272 c^{5} x^{5}-256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-88 c^{3} x^{3}+160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c x -32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \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 ) d}{1024 \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 ) d}{1024 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+128 c^{9} x^{9}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+272 c^{5} x^{5}+32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-88 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+8 c x \right ) \left (1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(783\) |
-1/8*a*x^3*(-c^2*d*x^2+d)^(5/2)/c^2/d-1/16*a/c^4*x*(-c^2*d*x^2+d)^(5/2)/d+ 1/64*a/c^4*x*(-c^2*d*x^2+d)^(3/2)+3/128*a/c^4*d*x*(-c^2*d*x^2+d)^(1/2)+3/1 28*a/c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b* (-3/256*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^5*arccosh(c*x )^2*d-1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x-1)^ (1/2)*(c*x+1)^(1/2)*c^8*x^8+272*c^5*x^5-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^ 6*x^6-88*c^3*x^3+160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(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+8*arccosh(c*x ))*d/(c*x+1)/c^5/(c*x-1)+1/1024*(-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))*d/(c*x+1)/c^5/ (c*x-1)+1/1024*(-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))*d/(c*x+1)/c^5/(c*x-1)-1/16384*( -d*(c^2*x^2-1))^(1/2)*(-128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+128*c^9*x^ 9+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-320*c^7*x^7-160*(c*x+1)^(1/2)*(c *x-1)^(1/2)*c^4*x^4+272*c^5*x^5+32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-88* c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)*(1+8*arccosh(c*x))*d/(c*x+1)/c^ 5/(c*x-1))
\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
integral(-(a*c^2*d*x^6 - a*d*x^4 + (b*c^2*d*x^6 - b*d*x^4)*arccosh(c*x))*s qrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
-1/128*(16*(-c^2*d*x^2 + d)^(5/2)*x^3/(c^2*d) - 2*(-c^2*d*x^2 + d)^(3/2)*x /c^4 + 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^4*d) - 3*sqrt(-c^2*d*x^2 + d)*d*x/c^4 - 3*d^(3/2)*arcsin(c*x)/c^5)*a + b*integrate((-c^2*d*x^2 + d)^(3/2)*x^4*l og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]