Integrand size = 29, antiderivative size = 336 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/d/x^3+1/3*b^2*c^2*(-c^2*d*x ^2+d)^(1/2)/x-1/3*b^2*c^3*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/ (c*x+1)^(1/2)-1/3*b*c*(-c^2*x^2+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2) /x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*c^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+ d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/3*b*c^3*(a+b*arccosh(c*x))*ln(1+1/( c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c* x+1)^(1/2)+1/3*b^2*c^3*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*( -c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.90 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=-\frac {d (1+c x) \left (a^2-a^2 c x-a^2 c^2 x^2-b^2 c^2 x^2+a^2 c^3 x^3+b^2 c^3 x^3-a b c x \sqrt {\frac {-1+c x}{1+c x}}-b^2 \left (-1+c x+c^2 x^2+c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \text {arccosh}(c x)^2-b \text {arccosh}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}-2 a (-1+c x)^2 (1+c x)+2 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-2 a b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log (c x)+b^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \]
-1/3*(d*(1 + c*x)*(a^2 - a^2*c*x - a^2*c^2*x^2 - b^2*c^2*x^2 + a^2*c^3*x^3 + b^2*c^3*x^3 - a*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - b^2*(-1 + c*x + c^2* x^2 + c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*x]^2 - b*ArcCos h[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*a*(-1 + c*x)^2*(1 + c*x) + 2* b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x])]) - 2*a*b *c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[c*x] + b^2*c^3*x^3*Sqrt[(-1 + c*x) /(1 + c*x)]*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/(x^3*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.63, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {6332, 25, 6327, 6335, 108, 27, 43, 6297, 25, 3042, 26, 4201, 2620, 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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6332 |
| \(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6335 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \int \frac {\sqrt {c x-1} \sqrt {c x+1}}{x^2}dx-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \left (\int \frac {c^2}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \left (c^2 \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {c^2 \int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {c^2 \int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {c^2 \int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(d*x^3) + (2*b*c*Sqrt[ d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/x^2 - (b*c*(-((S qrt[-1 + c*x]*Sqrt[1 + c*x])/x) + c*ArcCosh[c*x]))/2 - (I*c^2*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*A rcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4)))/b))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.77.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && 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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 , 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcCosh[c *x])/(f*(m + 1))), x] + (-Simp[b*c*((-d)^p/(f*(m + 1))) Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x] - Simp[2*e*(p/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x]), x], x]) / ; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[( m + 1)/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1750\) vs. \(2(314)=628\).
Time = 1.22 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.21
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1751\) |
| parts | \(\text {Expression too large to display}\) | \(1751\) |
1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x*arccosh(c*x)*c^4- 1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x- 1)^(1/2)*c^3-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3*ar ccosh(c*x)*c^6+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3* c^6-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c*x+1)/(c* x-1)*c^6+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c*x+1)/ (c*x-1)*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c*x+ 1)/(c*x-1)*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/ (c*x+1)/(c*x-1)*arccosh(c*x)^2-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2 )/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3 +2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1 )*c^8-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c*x+1)^(1/2) /(c*x-1)^(1/2)*c^7+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c*x +1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^3+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4 *x^4-3*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-1/3*b^2*(-d*(c^2*x^2 -1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x )^2*c^3+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(2*(c*x+1)^(1/2)*arccosh(c*x)*(c*x- 1)^(1/2)*c^2*x^2+2*c^3*x^3*arccosh(c*x)-2*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^ (1/2))^2)*x^3*c^3-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*x)/(c*x-1)^ (1/2)/(c*x+1)^(1/2)/x^3-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(3/2)-3*b^2*(-d*(c...
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
1/3*(c^4*d^2*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + I*(-1)^(-2*c^2*d*x^2 + 2* d)*c^2*d^(3/2)*log(-2*c^2*d + 2*d/x^2) + sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d )*d/x^2)*a*b*c/d + 1/3*b^2*((c^2*sqrt(d)*x^2 - sqrt(d))*sqrt(c*x + 1)*sqrt (-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^3 - 3*integrate(2/3* ((c*x + 1)*sqrt(c*x - 1)*c^2*sqrt(d)*x + (c^3*sqrt(d)*x^2 - c*sqrt(d))*sqr t(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c*x^4 + sqrt(c*x + 1)*sqrt(c*x - 1)*x^3), x)) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*b*ar ccosh(c*x)/(d*x^3) - 1/3*(-c^2*d*x^2 + d)^(3/2)*a^2/(d*x^3)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \]