\(\int \frac {(c-a^2 c x^2)^3}{\text {arccosh}(a x)} \, dx\) [18]

Optimal result

Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\frac {35 c^3 \text {Shi}(\text {arccosh}(a x))}{64 a}-\frac {21 c^3 \text {Shi}(3 \text {arccosh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(5 \text {arccosh}(a x))}{64 a}-\frac {c^3 \text {Shi}(7 \text {arccosh}(a x))}{64 a} \] Output:

35/64*c^3*Shi(arccosh(a*x))/a-21/64*c^3*Shi(3*arccosh(a*x))/a+7/64*c^3*Shi 
(5*arccosh(a*x))/a-1/64*c^3*Shi(7*arccosh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\frac {c^3 (35 \text {Shi}(\text {arccosh}(a x))-21 \text {Shi}(3 \text {arccosh}(a x))+7 \text {Shi}(5 \text {arccosh}(a x))-\text {Shi}(7 \text {arccosh}(a x)))}{64 a} \] Input:

Integrate[(c - a^2*c*x^2)^3/ArcCosh[a*x],x]
 

Output:

(c^3*(35*SinhIntegral[ArcCosh[a*x]] - 21*SinhIntegral[3*ArcCosh[a*x]] + 7* 
SinhIntegral[5*ArcCosh[a*x]] - SinhIntegral[7*ArcCosh[a*x]]))/(64*a)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91, 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.

\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx\)

\(\Big \downarrow \) 6321

\(\displaystyle -\frac {c^3 \int \frac {\left (\frac {a x-1}{a x+1}\right )^{7/2} (a x+1)^7}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c^3 \int \frac {i \sin (i \text {arccosh}(a x))^7}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i c^3 \int \frac {\sin (i \text {arccosh}(a x))^7}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {i c^3 \int \left (\frac {35 i \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{64 \text {arccosh}(a x)}-\frac {21 i \sinh (3 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}+\frac {7 i \sinh (5 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}-\frac {i \sinh (7 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i c^3 \left (\frac {35}{64} i \text {Shi}(\text {arccosh}(a x))-\frac {21}{64} i \text {Shi}(3 \text {arccosh}(a x))+\frac {7}{64} i \text {Shi}(5 \text {arccosh}(a x))-\frac {1}{64} i \text {Shi}(7 \text {arccosh}(a x))\right )}{a}\)

Input:

Int[(c - a^2*c*x^2)^3/ArcCosh[a*x],x]
 

Output:

((-I)*c^3*(((35*I)/64)*SinhIntegral[ArcCosh[a*x]] - ((21*I)/64)*SinhIntegr 
al[3*ArcCosh[a*x]] + ((7*I)/64)*SinhIntegral[5*ArcCosh[a*x]] - (I/64)*Sinh 
Integral[7*ArcCosh[a*x]]))/a
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

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]))
 

rule 6321

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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-21 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+7 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a}\)	\(44\)
default	\(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-21 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+7 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a}\)	\(44\)

Input:

int((-a^2*c*x^2+c)^3/arccosh(a*x),x,method=_RETURNVERBOSE)
 

Output:

1/64/a*c^3*(35*Shi(arccosh(a*x))-21*Shi(3*arccosh(a*x))+7*Shi(5*arccosh(a* 
x))-Shi(7*arccosh(a*x)))
 

Fricas [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x),x, algorithm="fricas")
 

Output:

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccosh(a*x) 
, x)
 

Sympy [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \] Input:

integrate((-a**2*c*x**2+c)**3/acosh(a*x),x)
 

Output:

-c**3*(Integral(3*a**2*x**2/acosh(a*x), x) + Integral(-3*a**4*x**4/acosh(a 
*x), x) + Integral(a**6*x**6/acosh(a*x), x) + Integral(-1/acosh(a*x), x))
 

Maxima [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x),x, algorithm="maxima")
 

Output:

-integrate((a^2*c*x^2 - c)^3/arccosh(a*x), x)
 

Giac [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x),x, algorithm="giac")
 

Output:

integrate(-(a^2*c*x^2 - c)^3/arccosh(a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{\mathrm {acosh}\left (a\,x\right )} \,d x \] Input:

int((c - a^2*c*x^2)^3/acosh(a*x),x)
 

Output:

int((c - a^2*c*x^2)^3/acosh(a*x), x)
 

Reduce [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=c^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acosh} \left (a x \right )}d x \right ) a^{6}+3 \left (\int \frac {x^{4}}{\mathit {acosh} \left (a x \right )}d x \right ) a^{4}-3 \left (\int \frac {x^{2}}{\mathit {acosh} \left (a x \right )}d x \right ) a^{2}+\int \frac {1}{\mathit {acosh} \left (a x \right )}d x \right ) \] Input:

int((-a^2*c*x^2+c)^3/acosh(a*x),x)
 

Output:

c**3*( - int(x**6/acosh(a*x),x)*a**6 + 3*int(x**4/acosh(a*x),x)*a**4 - 3*i 
nt(x**2/acosh(a*x),x)*a**2 + int(1/acosh(a*x),x))