Integrand size = 23, antiderivative size = 27 \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a x))}{2 a^3}-\frac {\log (\text {arcsinh}(a x))}{2 a^3} \] Output:
1/2*Chi(2*arcsinh(a*x))/a^3-1/2*ln(arcsinh(a*x))/a^3
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a x))-\log (\text {arcsinh}(a x))}{2 a^3} \] Input:
Integrate[x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]
Output:
(CoshIntegral[2*ArcSinh[a*x]] - Log[ArcSinh[a*x]])/(2*a^3)
Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6234, 3042, 25, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int \frac {a^2 x^2}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\frac {\sin (i \text {arcsinh}(a x))^2}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\sin (i \text {arcsinh}(a x))^2}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\int \left (\frac {1}{2 \text {arcsinh}(a x)}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \text {Chi}(2 \text {arcsinh}(a x))-\frac {1}{2} \log (\text {arcsinh}(a x))}{a^3}\) |
Input:
Int[x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]
Output:
(CoshIntegral[2*ArcSinh[a*x]]/2 - Log[ArcSinh[a*x]]/2)/a^3
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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*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 + 2, 0] && IGtQ[m, 0]
Time = 1.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\ln \left (\operatorname {arcsinh}\left (x a \right )\right )-\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (x a \right )\right )}{2 a^{3}}\) | \(21\) |
Input:
int(x^2/(a^2*x^2+1)^(1/2)/arcsinh(x*a),x,method=_RETURNVERBOSE)
Output:
-1/2*(ln(arcsinh(x*a))-Chi(2*arcsinh(x*a)))/a^3
\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(a^2*x^2+1)^(1/2)/arcsinh(a*x),x, algorithm="fricas")
Output:
integral(x^2/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/(a**2*x**2+1)**(1/2)/asinh(a*x),x)
Output:
Integral(x**2/(sqrt(a**2*x**2 + 1)*asinh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(a^2*x^2+1)^(1/2)/arcsinh(a*x),x, algorithm="maxima")
Output:
integrate(x^2/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(a^2*x^2+1)^(1/2)/arcsinh(a*x),x, algorithm="giac")
Output:
integrate(x^2/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^2}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \] Input:
int(x^2/(asinh(a*x)*(a^2*x^2 + 1)^(1/2)),x)
Output:
int(x^2/(asinh(a*x)*(a^2*x^2 + 1)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {a^{2} x^{2}+1}\, \mathit {asinh} \left (a x \right )}d x \] Input:
int(x^2/(a^2*x^2+1)^(1/2)/asinh(a*x),x)
Output:
int(x**2/(sqrt(a**2*x**2 + 1)*asinh(a*x)),x)