Integrand size = 8, antiderivative size = 95 \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}-\frac {x^2}{3 \text {arcsinh}(a x)^2}-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)}+\frac {2 \text {Chi}(2 \text {arcsinh}(a x))}{3 a^2} \]
-1/6/a^2/arcsinh(a*x)^2-1/3*x^2/arcsinh(a*x)^2+2/3*Chi(2*arcsinh(a*x))/a^2 -1/3*x*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^3-2/3*x*(a^2*x^2+1)^(1/2)/a/arcsin h(a*x)
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=-\frac {2 a x \sqrt {1+a^2 x^2}+\left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)+4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2-4 \text {arcsinh}(a x)^3 \text {Chi}(2 \text {arcsinh}(a x))}{6 a^2 \text {arcsinh}(a x)^3} \]
-1/6*(2*a*x*Sqrt[1 + a^2*x^2] + (1 + 2*a^2*x^2)*ArcSinh[a*x] + 4*a*x*Sqrt[ 1 + a^2*x^2]*ArcSinh[a*x]^2 - 4*ArcSinh[a*x]^3*CoshIntegral[2*ArcSinh[a*x] ])/(a^2*ArcSinh[a*x]^3)
Time = 0.69 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6194, 6198, 6233, 6193, 3042, 3782}
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.
| \(\displaystyle \int \frac {x}{\text {arcsinh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx}{3 a}+\frac {2}{3} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {2}{3} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\int \frac {x}{\text {arcsinh}(a x)^2}dx}{a}-\frac {x^2}{2 a \text {arcsinh}(a x)^2}\right )-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {\int \frac {\cosh (2 \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}-\frac {x \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{a}-\frac {x^2}{2 a \text {arcsinh}(a x)^2}\right )-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} a \left (-\frac {x^2}{2 a \text {arcsinh}(a x)^2}+\frac {-\frac {x \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}+\frac {\int \frac {\sin \left (2 i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}}{a}\right )-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {\text {Chi}(2 \text {arcsinh}(a x))}{a^2}-\frac {x \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{a}-\frac {x^2}{2 a \text {arcsinh}(a x)^2}\right )-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}\) |
-1/3*(x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^3) - 1/(6*a^2*ArcSinh[a*x]^2) + (2*a*(-1/2*x^2/(a*ArcSinh[a*x]^2) + (-((x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a *x])) + CoshIntegral[2*ArcSinh[a*x]]/a^2)/a))/3
3.1.70.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
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_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3 \,\operatorname {arcsinh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
| default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3 \,\operatorname {arcsinh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
1/a^2*(-1/6/arcsinh(a*x)^3*sinh(2*arcsinh(a*x))-1/6/arcsinh(a*x)^2*cosh(2* arcsinh(a*x))-1/3/arcsinh(a*x)*sinh(2*arcsinh(a*x))+2/3*Chi(2*arcsinh(a*x) ))
\[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x}{\operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^12*x^12 + 10*a^10*x^10 + 20*a^8*x^8 + 20*a^6*x^6 + 10*a^4*x^4 + 2*a^2*x^2 + 2*(a^7*x^7 + a^5*x^5)*(a^2*x^2 + 1)^(5/2) + 2*(5*a^8*x^8 + 9*a ^6*x^6 + 4*a^4*x^4)*(a^2*x^2 + 1)^2 + (4*a^12*x^12 + 20*a^10*x^10 + 40*a^8 *x^8 + 40*a^6*x^6 + 20*a^4*x^4 + 4*a^2*x^2 + 4*(a^7*x^7 + a^5*x^5)*(a^2*x^ 2 + 1)^(5/2) + (20*a^8*x^8 + 36*a^6*x^6 + 16*a^4*x^4 - 3*a^2*x^2 - 3)*(a^2 *x^2 + 1)^2 + (40*a^9*x^9 + 104*a^7*x^7 + 88*a^5*x^5 + 21*a^3*x^3 - 3*a*x) *(a^2*x^2 + 1)^(3/2) + (40*a^10*x^10 + 136*a^8*x^8 + 168*a^6*x^6 + 91*a^4* x^4 + 22*a^2*x^2 + 3)*(a^2*x^2 + 1) + (20*a^11*x^11 + 84*a^9*x^9 + 136*a^7 *x^7 + 107*a^5*x^5 + 42*a^3*x^3 + 7*a*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt (a^2*x^2 + 1))^2 + 4*(5*a^9*x^9 + 13*a^7*x^7 + 11*a^5*x^5 + 3*a^3*x^3)*(a^ 2*x^2 + 1)^(3/2) + 4*(5*a^10*x^10 + 17*a^8*x^8 + 21*a^6*x^6 + 11*a^4*x^4 + 2*a^2*x^2)*(a^2*x^2 + 1) + (2*a^12*x^12 + 10*a^10*x^10 + 20*a^8*x^8 + 20* a^6*x^6 + 10*a^4*x^4 + 2*a^2*x^2 + 2*(a^7*x^7 + a^5*x^5)*(a^2*x^2 + 1)^(5/ 2) + (10*a^8*x^8 + 18*a^6*x^6 + 9*a^4*x^4 + a^2*x^2)*(a^2*x^2 + 1)^2 + (20 *a^9*x^9 + 52*a^7*x^7 + 47*a^5*x^5 + 17*a^3*x^3 + 2*a*x)*(a^2*x^2 + 1)^(3/ 2) + (20*a^10*x^10 + 68*a^8*x^8 + 87*a^6*x^6 + 51*a^4*x^4 + 13*a^2*x^2 + 1 )*(a^2*x^2 + 1) + (10*a^11*x^11 + 42*a^9*x^9 + 69*a^7*x^7 + 55*a^5*x^5 + 2 1*a^3*x^3 + 3*a*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + 2*(5* a^11*x^11 + 21*a^9*x^9 + 34*a^7*x^7 + 26*a^5*x^5 + 9*a^3*x^3 + a*x)*sqrt(a ^2*x^2 + 1))/((a^12*x^10 + 5*a^10*x^8 + (a^2*x^2 + 1)^(5/2)*a^7*x^5 + 1...
\[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]