Integrand size = 26, antiderivative size = 454 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {15 d^2 \sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 \sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 \sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*d*x^2+d)^(5/2)/b/c/(a+b*arccosh(c*x))-1 5/16*d^2*(-c^2*d*x^2+d)^(1/2)*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)/b^2/ c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/4*d^2*(-c^2*d*x^2+d)^(1/2)*Chi(4*(a+b*arcc osh(c*x))/b)*sinh(4*a/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*d^2*(-c^2* d*x^2+d)^(1/2)*Chi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)/b^2/c/(c*x-1)^(1/2) /(c*x+1)^(1/2)+15/16*d^2*(-c^2*d*x^2+d)^(1/2)*cosh(2*a/b)*Shi(2*(a+b*arcco sh(c*x))/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/4*d^2*(-c^2*d*x^2+d)^(1/2) *cosh(4*a/b)*Shi(4*(a+b*arccosh(c*x))/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2) +3/16*d^2*(-c^2*d*x^2+d)^(1/2)*cosh(6*a/b)*Shi(6*(a+b*arccosh(c*x))/b)/b^2 /c/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d^4 \sqrt {-1+c x} \sqrt {1+c x} \left (-\frac {16 b \left (-1+c^2 x^2\right )^4}{a+b \text {arccosh}(c x)}+12 (-1+c x) (1+c x) \left (-2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+\text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )+3 (-1+c x) (1+c x) \left (3 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )\right )}{16 b^2 c \left (d-c^2 d x^2\right )^{3/2}} \] Input:
Integrate[(d - c^2*d*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2,x]
Output:
(d^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((-16*b*(-1 + c^2*x^2)^4)/(a + b*ArcCosh [c*x]) + 12*(-1 + c*x)*(1 + c*x)*(-2*CoshIntegral[2*(a/b + ArcCosh[c*x])]* Sinh[(2*a)/b] + CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 2*Cos h[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Cosh[(4*a)/b]*SinhIntegr al[4*(a/b + ArcCosh[c*x])]) + 3*(-1 + c*x)*(1 + c*x)*(3*CoshIntegral[2*(a/ b + ArcCosh[c*x])]*Sinh[(2*a)/b] - CoshIntegral[6*(a/b + ArcCosh[c*x])]*Si nh[(6*a)/b] - 3*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + Cosh[ (6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])])))/(16*b^2*c*(d - c^2*d*x^2) ^(3/2))
Time = 1.03 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.54, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6319, 6327, 6367, 25, 5971, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) |
\(\Big \downarrow \) 6319 |
\(\displaystyle \frac {6 c d^2 \sqrt {d-c^2 d x^2} \int \frac {x (1-c x)^2 (c x+1)^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 6327 |
\(\displaystyle \frac {6 c d^2 \sqrt {d-c^2 d x^2} \int \frac {x \left (1-c^2 x^2\right )^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 6367 |
\(\displaystyle \frac {6 d^2 \sqrt {d-c^2 d x^2} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {6 d^2 \sqrt {d-c^2 d x^2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {6 d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}+\frac {5 \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6 d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+\frac {5}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}\) |
Input:
Int[(d - c^2*d*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2,x]
Output:
-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d - c^2*d*x^2)^(5/2))/(b*c*(a + b*ArcCosh [c*x]))) + (6*d^2*Sqrt[d - c^2*d*x^2]*((-5*CoshIntegral[(2*(a + b*ArcCosh[ c*x]))/b]*Sinh[(2*a)/b])/32 + (CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Si nh[(4*a)/b])/8 - (CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b]*Sinh[(6*a)/b])/ 32 + (5*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/32 - (Cosh [(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8 + (Cosh[(6*a)/b]*Sin hIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32))/(b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x )^p*(-1 + c*x)^p)] Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.27 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.38
method | result | size |
default | \(-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{6} x^{6}-32 b \,c^{7} x^{7}+96 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{4} x^{4}+96 b \,c^{5} x^{5}-96 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}-96 b \,c^{3} x^{3}+12 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-15 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-12 \,\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+15 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+32 \sqrt {c x -1}\, \sqrt {c x +1}\, b +12 a \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-15 a \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-3 a \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} a -12 \,\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a +15 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a +32 b c x \right ) d^{2}}{32 \left (c x -1\right ) \left (c x +1\right ) c \,b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(628\) |
Input:
int((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
-1/32*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)* (-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^6*x^6-32*b*c^7*x^7+96*(c*x-1)^(1/2)*( c*x+1)^(1/2)*b*c^4*x^4+96*b*c^5*x^5-96*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x ^2-96*b*c^3*x^3+12*arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arc cosh(c*x)+4*a)/b)-15*arccosh(c*x)*b*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*a rccosh(c*x)+2*a)/b)-3*arccosh(c*x)*b*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b* arccosh(c*x)+6*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a) /b)*b*arccosh(c*x)-12*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/ b)*b*arccosh(c*x)+15*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b )*b*arccosh(c*x)+32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b+12*a*Ei(1,-4*arccosh(c*x )-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)-15*a*Ei(1,-2*arccosh(c*x)-2*a/b)*ex p(-(-b*arccosh(c*x)+2*a)/b)-3*a*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b*arcco sh(c*x)+6*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)*a -12*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)*a+15*Ei(1,2*arc cosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a+32*b*c*x)*d^2/(c*x-1)/(c*x+ 1)/c/b^2/(a+b*arccosh(c*x))
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)/(b^2*arc cosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\text {Timed out} \] Input:
integrate((-c**2*d*x**2+d)**(5/2)/(a+b*acosh(c*x))**2,x)
Output:
Timed out
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
-((c^6*d^(5/2)*x^6 - 3*c^4*d^(5/2)*x^4 + 3*c^2*d^(5/2)*x^2 - d^(5/2))*(c*x + 1)*sqrt(c*x - 1) + (c^7*d^(5/2)*x^7 - 3*c^5*d^(5/2)*x^5 + 3*c^3*d^(5/2) *x^3 - c*d^(5/2)*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c *x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integ rate(((6*c^6*d^(5/2)*x^6 - 11*c^4*d^(5/2)*x^4 + 4*c^2*d^(5/2)*x^2 + d^(5/2 ))*(c*x + 1)^(3/2)*(c*x - 1) + 6*(2*c^7*d^(5/2)*x^7 - 5*c^5*d^(5/2)*x^5 + 4*c^3*d^(5/2)*x^3 - c*d^(5/2)*x)*(c*x + 1)*sqrt(c*x - 1) + (6*c^8*d^(5/2)* x^8 - 19*c^6*d^(5/2)*x^6 + 21*c^4*d^(5/2)*x^4 - 9*c^2*d^(5/2)*x^2 + d^(5/2 ))*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^4*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^ 2*x^2 - 2*a*b*c^2*x^2 + 2*(a*b*c^3*x^3 - a*b*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + a*b + (b^2*c^4*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^2*x^2 - 2*b^2*c^2*x^2 + 2*(b^2*c^3*x^3 - b^2*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + b^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")
Output:
integrate((-c^2*d*x^2 + d)^(5/2)/(b*arccosh(c*x) + a)^2, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((d - c^2*d*x^2)^(5/2)/(a + b*acosh(c*x))^2,x)
Output:
int((d - c^2*d*x^2)^(5/2)/(a + b*acosh(c*x))^2, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\sqrt {d}\, d^{2} \left (\int \frac {\sqrt {-c^{2} x^{2}+1}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x +\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{4}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{2}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}\right ) \] Input:
int((-c^2*d*x^2+d)^(5/2)/(a+b*acosh(c*x))^2,x)
Output:
sqrt(d)*d**2*(int(sqrt( - c**2*x**2 + 1)/(acosh(c*x)**2*b**2 + 2*acosh(c*x )*a*b + a**2),x) + int((sqrt( - c**2*x**2 + 1)*x**4)/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**4 - 2*int((sqrt( - c**2*x**2 + 1)*x**2)/(a cosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**2)