Integrand size = 21, antiderivative size = 64 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {6 x}{a}+\frac {6 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2}-\frac {3 x \text {arcsinh}(a x)^2}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a^2} \] Output:
-6*x/a+6*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^2-3*x*arcsinh(a*x)^2/a+(a^2*x^2+ 1)^(1/2)*arcsinh(a*x)^3/a^2
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {-6 a x+6 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)-3 a x \text {arcsinh}(a x)^2+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a^2} \] Input:
Integrate[(x*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]
Output:
(-6*a*x + 6*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 3*a*x*ArcSinh[a*x]^2 + Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a^2
Time = 0.40 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6213, 6187, 6213, 24}
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.
\(\displaystyle \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \int \text {arcsinh}(a x)^2dx}{a}\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \int \frac {x \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{a}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )\right )}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a}\) |
Input:
Int[(x*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]
Output:
(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a^2 - (3*(x*ArcSinh[a*x]^2 - 2*a*(-(x/a ) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2)))/a
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Time = 1.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (x a \right )^{3} a^{2} x^{2}+\operatorname {arcsinh}\left (x a \right )^{3}-3 \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}\, x a +6 \,\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}+6 \,\operatorname {arcsinh}\left (x a \right )-6 x a \sqrt {a^{2} x^{2}+1}}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(90\) |
orering | \(\frac {\left (a^{4} x^{4}+8 a^{2} x^{2}+8\right ) \operatorname {arcsinh}\left (x a \right )^{3}}{a^{4} x^{2} \sqrt {a^{2} x^{2}+1}}-\frac {\left (a^{4} x^{4}+6 a^{2} x^{2}+8\right ) \left (\frac {\operatorname {arcsinh}\left (x a \right )^{3}}{\sqrt {a^{2} x^{2}+1}}+\frac {3 x \operatorname {arcsinh}\left (x a \right )^{2} a}{a^{2} x^{2}+1}-\frac {x^{2} \operatorname {arcsinh}\left (x a \right )^{3} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{x^{2} a^{4}}-\frac {2 \left (a^{2} x^{2}+1\right ) \left (a^{2} x^{2}-2\right ) \left (\frac {6 \operatorname {arcsinh}\left (x a \right )^{2} a}{a^{2} x^{2}+1}-\frac {3 \operatorname {arcsinh}\left (x a \right )^{3} a^{2} x}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 \,\operatorname {arcsinh}\left (x a \right ) x \,a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {9 x^{2} \operatorname {arcsinh}\left (x a \right )^{2} a^{3}}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{3} \operatorname {arcsinh}\left (x a \right )^{3} a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{4} x}-\frac {\left (a^{2} x^{2}+1\right )^{2} \left (\frac {18 \,\operatorname {arcsinh}\left (x a \right ) a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {39 \operatorname {arcsinh}\left (x a \right )^{2} a^{3} x}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {18 \operatorname {arcsinh}\left (x a \right )^{3} a^{4} x^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \operatorname {arcsinh}\left (x a \right )^{3} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 a^{3} x}{\left (a^{2} x^{2}+1\right )^{2}}-\frac {36 \,\operatorname {arcsinh}\left (x a \right ) x^{2} a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {45 x^{3} \operatorname {arcsinh}\left (x a \right )^{2} a^{5}}{\left (a^{2} x^{2}+1\right )^{3}}-\frac {15 x^{4} \operatorname {arcsinh}\left (x a \right )^{3} a^{6}}{\left (a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{a^{4}}\) | \(471\) |
Input:
int(x*arcsinh(x*a)^3/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/a^2/(a^2*x^2+1)^(1/2)*(arcsinh(x*a)^3*a^2*x^2+arcsinh(x*a)^3-3*arcsinh(x *a)^2*(a^2*x^2+1)^(1/2)*x*a+6*arcsinh(x*a)*x^2*a^2+6*arcsinh(x*a)-6*x*a*(a ^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.44 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 6 \, a x - 6 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \] Input:
integrate(x*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
-(3*a*x*log(a*x + sqrt(a^2*x^2 + 1))^2 - sqrt(a^2*x^2 + 1)*log(a*x + sqrt( a^2*x^2 + 1))^3 + 6*a*x - 6*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)) )/a^2
Time = 0.35 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {3 x \operatorname {asinh}^{2}{\left (a x \right )}}{a} - \frac {6 x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a^{2}} + \frac {6 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)
Output:
Piecewise((-3*x*asinh(a*x)**2/a - 6*x/a + sqrt(a**2*x**2 + 1)*asinh(a*x)** 3/a**2 + 6*sqrt(a**2*x**2 + 1)*asinh(a*x)/a**2, Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 \, x \operatorname {arsinh}\left (a x\right )^{2}}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}}{a^{2}} - \frac {6 \, {\left (x - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
-3*x*arcsinh(a*x)^2/a + sqrt(a^2*x^2 + 1)*arcsinh(a*x)^3/a^2 - 6*(x - sqrt (a^2*x^2 + 1)*arcsinh(a*x)/a)/a
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.58 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac {3 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a} \] Input:
integrate(x*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^3/a^2 - 3*(x*log(a*x + sqrt (a^2*x^2 + 1))^2 + 2*a*(x/a - sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1 ))/a^2))/a
Timed out. \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {a^2\,x^2+1}} \,d x \] Input:
int((x*asinh(a*x)^3)/(a^2*x^2 + 1)^(1/2),x)
Output:
int((x*asinh(a*x)^3)/(a^2*x^2 + 1)^(1/2), x)
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2}+1}\, \mathit {asinh} \left (a x \right )^{3}-3 \mathit {asinh} \left (a x \right )^{2} a x +6 \sqrt {a^{2} x^{2}+1}\, \mathit {asinh} \left (a x \right )-6 a x}{a^{2}} \] Input:
int(x*asinh(a*x)^3/(a^2*x^2+1)^(1/2),x)
Output:
(sqrt(a**2*x**2 + 1)*asinh(a*x)**3 - 3*asinh(a*x)**2*a*x + 6*sqrt(a**2*x** 2 + 1)*asinh(a*x) - 6*a*x)/a**2