\(\int \frac {(c+a^2 c x^2)^3}{\text {arcsinh}(a x)} \, dx\) [17]

Optimal result

Integrand size = 19, antiderivative size = 67 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\frac {35 c^3 \text {Chi}(\text {arcsinh}(a x))}{64 a}+\frac {21 c^3 \text {Chi}(3 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Chi}(5 \text {arcsinh}(a x))}{64 a}+\frac {c^3 \text {Chi}(7 \text {arcsinh}(a x))}{64 a} \] Output:

35/64*c^3*Chi(arcsinh(a*x))/a+21/64*c^3*Chi(3*arcsinh(a*x))/a+7/64*c^3*Chi 
(5*arcsinh(a*x))/a+1/64*c^3*Chi(7*arcsinh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\frac {c^3 (35 \text {Chi}(\text {arcsinh}(a x))+21 \text {Chi}(3 \text {arcsinh}(a x))+7 \text {Chi}(5 \text {arcsinh}(a x))+\text {Chi}(7 \text {arcsinh}(a x)))}{64 a} \] Input:

Integrate[(c + a^2*c*x^2)^3/ArcSinh[a*x],x]
 

Output:

(c^3*(35*CoshIntegral[ArcSinh[a*x]] + 21*CoshIntegral[3*ArcSinh[a*x]] + 7* 
CoshIntegral[5*ArcSinh[a*x]] + CoshIntegral[7*ArcSinh[a*x]]))/(64*a)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6206, 3042, 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.

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

\(\Big \downarrow \) 6206

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c^3 \int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )^7}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {c^3 \int \left (\frac {21 \cosh (3 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {7 \cosh (5 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {\cosh (7 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {35 \sqrt {a^2 x^2+1}}{64 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c^3 \left (\frac {35}{64} \text {Chi}(\text {arcsinh}(a x))+\frac {21}{64} \text {Chi}(3 \text {arcsinh}(a x))+\frac {7}{64} \text {Chi}(5 \text {arcsinh}(a x))+\frac {1}{64} \text {Chi}(7 \text {arcsinh}(a x))\right )}{a}\)

Input:

Int[(c + a^2*c*x^2)^3/ArcSinh[a*x],x]
 

Output:

(c^3*((35*CoshIntegral[ArcSinh[a*x]])/64 + (21*CoshIntegral[3*ArcSinh[a*x] 
])/64 + (7*CoshIntegral[5*ArcSinh[a*x]])/64 + CoshIntegral[7*ArcSinh[a*x]] 
/64))/a
 

Defintions of rubi rules used

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 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
 

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (x a \right )\right )+21 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )+7 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )+\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (x a \right )\right )\right )}{64 a}\)	\(42\)
default	\(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (x a \right )\right )+21 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )+7 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )+\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (x a \right )\right )\right )}{64 a}\)	\(42\)

Input:

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

Output:

1/64/a*c^3*(35*Chi(arcsinh(x*a))+21*Chi(3*arcsinh(x*a))+7*Chi(5*arcsinh(x* 
a))+Chi(7*arcsinh(x*a)))
 

Fricas [F]

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

integrate((a^2*c*x^2+c)^3/arcsinh(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)/arcsinh(a*x), 
 x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate((a^2*c*x^2 + c)^3/arcsinh(a*x), x)
 

Giac [F]

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

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

Output:

integrate((a^2*c*x^2 + c)^3/arcsinh(a*x), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c**3*(int(x**6/asinh(a*x),x)*a**6 + 3*int(x**4/asinh(a*x),x)*a**4 + 3*int( 
x**2/asinh(a*x),x)*a**2 + int(1/asinh(a*x),x))