Integrand size = 20, antiderivative size = 50 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\frac {5 c^2 \text {Shi}(\text {arccosh}(a x))}{8 a}-\frac {5 c^2 \text {Shi}(3 \text {arccosh}(a x))}{16 a}+\frac {c^2 \text {Shi}(5 \text {arccosh}(a x))}{16 a} \] Output:
5/8*c^2*Shi(arccosh(a*x))/a-5/16*c^2*Shi(3*arccosh(a*x))/a+1/16*c^2*Shi(5* arccosh(a*x))/a
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\frac {c^2 (10 \text {Shi}(\text {arccosh}(a x))-5 \text {Shi}(3 \text {arccosh}(a x))+\text {Shi}(5 \text {arccosh}(a x)))}{16 a} \] Input:
Integrate[(c - a^2*c*x^2)^2/ArcCosh[a*x],x]
Output:
(c^2*(10*SinhIntegral[ArcCosh[a*x]] - 5*SinhIntegral[3*ArcCosh[a*x]] + Sin hIntegral[5*ArcCosh[a*x]]))/(16*a)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6321, 3042, 26, 3793, 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.
\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx\) |
\(\Big \downarrow \) 6321 |
\(\displaystyle \frac {c^2 \int \frac {\left (\frac {a x-1}{a x+1}\right )^{5/2} (a x+1)^5}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {c^2 \int -\frac {i \sin (i \text {arccosh}(a x))^5}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i c^2 \int \frac {\sin (i \text {arccosh}(a x))^5}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {i c^2 \int \left (\frac {5 i \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{8 \text {arccosh}(a x)}-\frac {5 i \sinh (3 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}+\frac {i \sinh (5 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i c^2 \left (\frac {5}{8} i \text {Shi}(\text {arccosh}(a x))-\frac {5}{16} i \text {Shi}(3 \text {arccosh}(a x))+\frac {1}{16} i \text {Shi}(5 \text {arccosh}(a x))\right )}{a}\) |
Input:
Int[(c - a^2*c*x^2)^2/ArcCosh[a*x],x]
Output:
((-I)*c^2*(((5*I)/8)*SinhIntegral[ArcCosh[a*x]] - ((5*I)/16)*SinhIntegral[ 3*ArcCosh[a*x]] + (I/16)*SinhIntegral[5*ArcCosh[a*x]]))/a
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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {c^{2} \left (10 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-5 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a}\) | \(33\) |
default | \(\frac {c^{2} \left (10 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-5 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a}\) | \(33\) |
Input:
int((-a^2*c*x^2+c)^2/arccosh(a*x),x,method=_RETURNVERBOSE)
Output:
1/16/a*c^2*(10*Shi(arccosh(a*x))-5*Shi(3*arccosh(a*x))+Shi(5*arccosh(a*x)) )
\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^2/arccosh(a*x),x, algorithm="fricas")
Output:
integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)/arccosh(a*x), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}{\left (a x \right )}}\, dx\right ) \] Input:
integrate((-a**2*c*x**2+c)**2/acosh(a*x),x)
Output:
c**2*(Integral(-2*a**2*x**2/acosh(a*x), x) + Integral(a**4*x**4/acosh(a*x) , x) + Integral(1/acosh(a*x), x))
\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^2/arccosh(a*x),x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 - c)^2/arccosh(a*x), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^2/arccosh(a*x),x, algorithm="giac")
Output:
integrate((a^2*c*x^2 - c)^2/arccosh(a*x), x)
Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \] Input:
int((c - a^2*c*x^2)^2/acosh(a*x),x)
Output:
int((c - a^2*c*x^2)^2/acosh(a*x), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)} \, dx=c^{2} \left (\left (\int \frac {x^{4}}{\mathit {acosh} \left (a x \right )}d x \right ) a^{4}-2 \left (\int \frac {x^{2}}{\mathit {acosh} \left (a x \right )}d x \right ) a^{2}+\int \frac {1}{\mathit {acosh} \left (a x \right )}d x \right ) \] Input:
int((-a^2*c*x^2+c)^2/acosh(a*x),x)
Output:
c**2*(int(x**4/acosh(a*x),x)*a**4 - 2*int(x**2/acosh(a*x),x)*a**2 + int(1/ acosh(a*x),x))