Integrand size = 27, antiderivative size = 224 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^5 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \] Output:
-1/6*b/c^5/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*x^3*(a +b*arccosh(c*x))/c^2/d/(-c^2*d*x^2+d)^(3/2)-x*(a+b*arccosh(c*x))/c^4/d^2/( -c^2*d*x^2+d)^(1/2)-1/2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c^5/d^ 3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(-c^2*x^ 2+1)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 0.64 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {2 a c x \left (-3+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{\left (-1+c^2 x^2\right )^2}-6 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b d \left (-8 c x \text {arccosh}(c x)-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+2 c x \text {arccosh}(c x)}{-1+c^2 x^2}+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (3 \text {arccosh}(c x)^2+8 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{\sqrt {d-c^2 d x^2}}}{6 c^5 d^3} \] Input:
Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]
Output:
((2*a*c*x*(-3 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)^2 - 6*a*Sqr t[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*d*(-8 *c*x*ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCosh[ c*x])/(-1 + c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*ArcCosh[c*x ]^2 + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/Sqrt[d - c^2*d*x^2])/ (6*c^5*d^3)
Time = 0.89 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6349, 82, 243, 49, 2009, 6349, 25, 82, 240, 6307}
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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^3}{(1-c x)^2 (c x+1)^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^3}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2}{\left (1-c^2 x^2\right )^2}dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {1}{c^2 \left (c^2 x^2-1\right )}+\frac {1}{c^2 \left (c^2 x^2-1\right )^2}\right )dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle -\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6307 |
| \(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]
Output:
(x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1/(c^4*(1 - c^2*x^2)) + Log[1 - c^2*x^2]/c^4))/(6*c*d^ 2*Sqrt[d - c^2*d*x^2]) - ((x*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x ^2]) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sq rt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2* c^3*d*Sqrt[d - c^2*d*x^2]))/(c^2*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x ] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
Time = 0.50 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(\frac {a \,x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-8 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}-6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x +16 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-c^{2} x^{2}+3 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )+1\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3} c^{5}}\) | \(366\) |
| parts | \(\frac {a \,x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-8 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}-6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x +16 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-c^{2} x^{2}+3 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )+1\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3} c^{5}}\) | \(366\) |
Input:
int(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*a*x^3/c^2/d/(-c^2*d*x^2+d)^(3/2)-a/c^4/d^2*x/(-c^2*d*x^2+d)^(1/2)+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))-1/6*b*(-d *(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*x^6-3*c^4*x^4+3*c^2*x ^2-1)/d^3/c^5*(3*arccosh(c*x)^2*x^4*c^4-8*arccosh(c*x)*(c*x+1)^(1/2)*(c*x- 1)^(1/2)*x^3*c^3-8*arccosh(c*x)*c^4*x^4+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1 /2))^2-1)*x^4*c^4-6*arccosh(c*x)^2*x^2*c^2+6*arccosh(c*x)*(c*x+1)^(1/2)*(c *x-1)^(1/2)*c*x+16*c^2*x^2*arccosh(c*x)-16*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^( 1/2))^2-1)*x^2*c^2-c^2*x^2+3*arccosh(c*x)^2-8*arccosh(c*x)+8*ln((c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))^2-1)+1)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas ")
Output:
integral(-(b*x^4*arccosh(c*x) + a*x^4)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**4*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral(x**4*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima ")
Output:
1/3*(x*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c ^4*d)) - x/(sqrt(-c^2*d*x^2 + d)*c^4*d^2) + 3*arcsin(c*x)/(c^5*d^(5/2)))*a + b*integrate(x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d) ^(5/2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)*x^4/(-c^2*d*x^2 + d)^(5/2), x)
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2),x)
Output:
int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{7} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{5}-4 a \,c^{3} x^{3}+3 a c x}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int(x^4*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(5/2),x)
Output:
(3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1) *asin(c*x)*a + 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**4)/(sqrt( - c** 2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2* x**2 + 1)),x)*b*c**7*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**4) /(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b*c**5 - 4*a*c**3*x**3 + 3*a*c*x)/(3*sqrt(d)*sq rt( - c**2*x**2 + 1)*c**5*d**2*(c**2*x**2 - 1))