Integrand size = 27, antiderivative size = 324 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c^3}{6 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 x^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 (a+b \text {arccosh}(c x))}{d^2 x^3 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d^3 x^3}-\frac {16 c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d^3 x}-\frac {8 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \] Output:
-1/6*b*c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/6*b*c*(c *x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x^2/(-c^2*d*x^2+d)^(1/2)+1/3*(a+b*arccosh(c* x))/d/x^3/(-c^2*d*x^2+d)^(3/2)+2*(a+b*arccosh(c*x))/d^2/x^3/(-c^2*d*x^2+d) ^(1/2)-8/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/d^3/x^3-16/3*c^2*(-c^2* d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/d^3/x-8/3*b*c^3*(c*x-1)^(1/2)*(c*x+1)^(1 /2)*ln(x)/d^2/(-c^2*d*x^2+d)^(1/2)-4/3*b*c^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*l n(-c^2*x^2+1)/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.73 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 a+12 a c^2 x^2-48 a c^4 x^4+32 a c^6 x^6-b c x \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \text {arccosh}(c x)-16 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \log (x)+8 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )-8 b c^5 x^5 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 d^2 x^3 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]
Output:
(2*a + 12*a*c^2*x^2 - 48*a*c^4*x^4 + 32*a*c^6*x^6 - b*c*x*Sqrt[-1 + c*x]*S qrt[1 + c*x] + 2*b*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcCosh[c*x] - 16*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*Log[x] + 8*b*c^ 3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2] - 8*b*c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(6*d^2*x^3*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 1.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 2331, 2123, 2009}
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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{x^3 \left (1-c^2 x^2\right )^2}dx}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{x^4 \left (1-c^2 x^2\right )^2}dx^2}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {8 c^4}{c^2 x^2-1}-\frac {c^4}{\left (c^2 x^2-1\right )^2}+\frac {8 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {c^2}{1-c^2 x^2}+8 c^2 \log \left (x^2\right )+8 c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}\right )}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]
Output:
-1/3*(a + b*ArcCosh[c*x])/(d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*Ar cCosh[c*x]))/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcCosh[c*x]))/ (3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[ d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*(-x^(-2) - c^2/(1 - c^2*x^2) + 8*c^2*Log[x^2] + 8*c^2*Log[1 - c^2*x^2]))/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c *x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo sh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c *x])] Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b , c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Time = 0.53 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(a \left (-\frac {1}{3 d \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+2 c^{2} \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+32 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{7} c^{7}-48 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-64 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+32 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{5} c^{5}+12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}-c^{3} x^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{3}}\) | \(403\) |
| parts | \(a \left (-\frac {1}{3 d \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+2 c^{2} \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+32 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{7} c^{7}-48 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-64 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+32 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{5} c^{5}+12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}-c^{3} x^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{3}}\) | \(403\) |
Input:
int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
a*(-1/3/d/x^3/(-c^2*d*x^2+d)^(3/2)+2*c^2*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4*c^ 2*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+2/3/d^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)*(32*(c*x-1)^(1/2)*(c*x+1) ^(1/2)*arccosh(c*x)*c^6*x^6+32*arccosh(c*x)*c^7*x^7-16*ln((c*x+(c*x-1)^(1/ 2)*(c*x+1)^(1/2))^4-1)*x^7*c^7-48*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x) *x^4*c^4-64*arccosh(c*x)*c^5*x^5+32*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4 -1)*x^5*c^5+12*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2+32*c^3*x^3 *arccosh(c*x)-16*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^3*c^3-c^3*x^3 +2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)/d^3/(c^6*x^6-3*c^4*x^4+3* c^2*x^2-1)/x^3
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas ")
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^10 - 3*c^4* d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**(5/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {8 \, c^{2} \sqrt {-d} \log \left (c x + 1\right )}{d^{3}} + \frac {8 \, c^{2} \sqrt {-d} \log \left (c x - 1\right )}{d^{3}} + \frac {16 \, c^{2} \sqrt {-d} \log \left (x\right )}{d^{3}} + \frac {\sqrt {-d}}{c^{2} d^{3} x^{4} - d^{3} x^{2}}\right )} + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} b \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima ")
Output:
1/6*b*c*(8*c^2*sqrt(-d)*log(c*x + 1)/d^3 + 8*c^2*sqrt(-d)*log(c*x - 1)/d^3 + 16*c^2*sqrt(-d)*log(x)/d^3 + sqrt(-d)/(c^2*d^3*x^4 - d^3*x^2)) + 1/3*(1 6*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*b*a rccosh(c*x) + 1/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d* x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*a
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^4), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(5/2)), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{8}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,c^{2} x^{5}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{8}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,x^{3}+16 a \,c^{6} x^{6}-24 a \,c^{4} x^{4}+6 a \,c^{2} x^{2}+a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} x^{3} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acosh(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x)
Output:
(3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**8 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**6 + sqrt( - c**2*x**2 + 1)*x**4),x)*b* c**2*x**5 - 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1 )*c**4*x**8 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**6 + sqrt( - c**2*x**2 + 1)* x**4),x)*b*x**3 + 16*a*c**6*x**6 - 24*a*c**4*x**4 + 6*a*c**2*x**2 + a)/(3* sqrt(d)*sqrt( - c**2*x**2 + 1)*d**2*x**3*(c**2*x**2 - 1))