Integrand size = 29, antiderivative size = 413 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a b x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 (1-c x) (1+c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \]
2*b^2*(-c*x+1)*(c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)+x^2*(a+b*arccosh(c*x))^2 /c^2/d/(-c^2*d*x^2+d)^(1/2)+4*a*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d/(-c^ 2*d*x^2+d)^(1/2)+4*b^2*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d/(- c^2*d*x^2+d)^(1/2)-2*b*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^ 3/d/(-c^2*d*x^2+d)^(1/2)+4*b*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^4/d/(-c^2*d*x^2+d)^(1/2)+2*b^ 2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/ c^4/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) )*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^4/d/(-c^2*d*x^2+d)^(1/2)+2*(a+b*arccosh(c* x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d^2
Time = 1.42 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 a^2 \left (-2+c^2 x^2\right )+2 a b \left (3 \text {arccosh}(c x)-\text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )+\sinh (2 \text {arccosh}(c x))\right )+b^2 \left (2+3 \text {arccosh}(c x)^2-2 \cosh (2 \text {arccosh}(c x))-\text {arccosh}(c x)^2 \cosh (2 \text {arccosh}(c x))-4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )-4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{2 c^4 d \sqrt {d-c^2 d x^2}} \]
(-2*a^2*(-2 + c^2*x^2) + 2*a*b*(3*ArcCosh[c*x] - ArcCosh[c*x]*Cosh[2*ArcCo sh[c*x]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[Cosh[ArcCosh[c*x]/2 ]] - Log[Sinh[ArcCosh[c*x]/2]]) + Sinh[2*ArcCosh[c*x]]) + b^2*(2 + 3*ArcCo sh[c*x]^2 - 2*Cosh[2*ArcCosh[c*x]] - ArcCosh[c*x]^2*Cosh[2*ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c *x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-Ar cCosh[c*x])] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcC osh[c*x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh [c*x])] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(2*c^4*d*Sqrt[d - c^2*d*x^ 2])
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.69, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {6349, 25, 6327, 6329, 2009, 6353, 83, 6318, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle -\frac {2 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {x (a+b \text {arccosh}(c x))}{c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
(x^2*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d)) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCosh[c*x]))/(c*S qrt[d - c^2*d*x^2])))/(c^2*d) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((b*Sqrt [-1 + c*x]*Sqrt[1 + c*x])/c^3 - (x*(a + b*ArcCosh[c*x]))/c^2 - (I*((2*I)*( a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c* x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/c^3))/(c*d*Sqrt[d - c^2*d*x^2])
3.3.6.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
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_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(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] && GtQ[n, 0] && NeQ[p, -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]
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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(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*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.20 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+2 c x \,\operatorname {arccosh}\left (c x \right )+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}+2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}+\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\) | \(648\) |
| parts | \(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+2 c x \,\operatorname {arccosh}\left (c x \right )+2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}+2 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}+\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\) | \(648\) |
a^2*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2))+b^2*(c* x-1)^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(arccosh(c*x)^2*(c*x+1)^(1 /2)*(c*x-1)^(1/2)*x^2*c^2-2*c^3*x^3*arccosh(c*x)-2*arccosh(c*x)*ln(1+c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)* (c*x+1)^(1/2))*x^2*c^2+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-2*polylog(2,- c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+2*polylog(2,c*x+(c*x-1)^(1/2)*(c* x+1)^(1/2))*x^2*c^2-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2+2*c*x*arc cosh(c*x)+2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*arccosh(c *x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2* polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*polylog(2,c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2)))/(c^2*x^2-1)^2/d^2/c^4+2*a*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(- d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-c^3 *x^3+ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2-ln(1+c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))*x^2*c^2-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-ln(( c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)+ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/( c^2*x^2-1)^2/d^2/c^4
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral((b^2*x^3*arccosh(c*x)^2 + 2*a*b*x^3*arccosh(c*x) + a^2*x^3)*sqrt( -c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-a*b*c*(2*sqrt(-d)*x/(c^4*d^2) + sqrt(-d)*log(c*x + 1)/(c^5*d^2) - sqrt(-d )*log(c*x - 1)/(c^5*d^2)) - 2*a*b*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(s qrt(-c^2*d*x^2 + d)*c^4*d))*arccosh(c*x) - a^2*(x^2/(sqrt(-c^2*d*x^2 + d)* c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d)) - b^2*((c^2*x^2 - 2)*log(c*x + sq rt(c*x + 1)*sqrt(c*x - 1))^2/(sqrt(c*x + 1)*sqrt(-c*x + 1)*c^4*d^(3/2)) - integrate(2*(c^4*x^4 - 3*c^2*x^2 + (c^3*x^3 - 2*c*x)*sqrt(c*x + 1)*sqrt(c* x - 1) + 2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(((c^5*d^(3/2)*x^2 - c^ 3*d^(3/2))*(c*x + 1)*sqrt(c*x - 1) + (c^6*d^(3/2)*x^3 - c^4*d^(3/2)*x)*sqr t(c*x + 1))*sqrt(-c*x + 1)), x))
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]