Integrand size = 21, antiderivative size = 86 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 x^2}{16 a^3}-\frac {x^4}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}+\frac {3 \text {arcsinh}(a x)^2}{16 a^5} \] Output:
3/16*x^2/a^3-1/16*x^4/a-3/8*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^4+1/4*x^3*( a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^2+3/16*arcsinh(a*x)^2/a^5
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 a^2 x^2-a^4 x^4+2 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)+3 \text {arcsinh}(a x)^2}{16 a^5} \] Input:
Integrate[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]
Output:
(3*a^2*x^2 - a^4*x^4 + 2*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a* x] + 3*ArcSinh[a*x]^2)/(16*a^5)
Time = 0.75 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6227, 15, 6227, 15, 6198}
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.
\(\displaystyle \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle -\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {\int x^3dx}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle -\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {3 \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\) |
Input:
Int[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]
Output:
-1/16*x^4/a + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) - (3*(-1/4*x^2/ a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)))/ (4*a^2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {4 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}-a^{4} x^{4}-6 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a +3 a^{2} x^{2}+3 \operatorname {arcsinh}\left (x a \right )^{2}+3}{16 a^{5}}\) | \(74\) |
Input:
int(x^4*arcsinh(x*a)/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/16*(4*x^3*a^3*arcsinh(x*a)*(a^2*x^2+1)^(1/2)-a^4*x^4-6*arcsinh(x*a)*(a^2 *x^2+1)^(1/2)*x*a+3*a^2*x^2+3*arcsinh(x*a)^2+3)/a^5
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^{4} x^{4} - 3 \, a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{16 \, a^{5}} \] Input:
integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
-1/16*(a^4*x^4 - 3*a^2*x^2 - 2*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a *x + sqrt(a^2*x^2 + 1)) - 3*log(a*x + sqrt(a^2*x^2 + 1))^2)/a^5
Time = 0.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{4}}{16 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{2}}{16 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{8 a^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**4*asinh(a*x)/(a**2*x**2+1)**(1/2),x)
Output:
Piecewise((-x**4/(16*a) + x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)/(4*a**2) + 3 *x**2/(16*a**3) - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(8*a**4) + 3*asinh(a* x)**2/(16*a**5), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {1}{16} \, {\left (\frac {x^{4}}{a^{2}} - \frac {3 \, x^{2}}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )^{2}}{a^{6}}\right )} a + \frac {1}{8} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} x^{3}}{a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} \operatorname {arsinh}\left (a x\right ) \] Input:
integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
-1/16*(x^4/a^2 - 3*x^2/a^4 + 3*arcsinh(a*x)^2/a^6)*a + 1/8*(2*sqrt(a^2*x^2 + 1)*x^3/a^2 - 3*sqrt(a^2*x^2 + 1)*x/a^4 + 3*arcsinh(a*x)/a^5)*arcsinh(a* x)
\[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^4*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(x^4*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)
Timed out. \[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^4\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \] Input:
int((x^4*asinh(a*x))/(a^2*x^2 + 1)^(1/2),x)
Output:
int((x^4*asinh(a*x))/(a^2*x^2 + 1)^(1/2), x)
\[ \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right ) x^{4}}{\sqrt {a^{2} x^{2}+1}}d x \] Input:
int(x^4*asinh(a*x)/(a^2*x^2+1)^(1/2),x)
Output:
int((asinh(a*x)*x**4)/sqrt(a**2*x**2 + 1),x)