Integrand size = 6, antiderivative size = 50 \[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=-\frac {\sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{2 a} \]
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=-\frac {\sqrt {1+a^2 x^2}+a x \text {arcsinh}(a x)-\text {arcsinh}(a x)^2 \text {Chi}(\text {arcsinh}(a x))}{2 a \text {arcsinh}(a x)^2} \]
-1/2*(Sqrt[1 + a^2*x^2] + a*x*ArcSinh[a*x] - ArcSinh[a*x]^2*CoshIntegral[A rcSinh[a*x]])/(a*ArcSinh[a*x]^2)
Time = 0.42 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6188, 6233, 6189, 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.
\(\displaystyle \int \frac {1}{\text {arcsinh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6188 |
\(\displaystyle \frac {1}{2} a \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx-\frac {\sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
\(\Big \downarrow \) 6233 |
\(\displaystyle \frac {1}{2} a \left (\frac {\int \frac {1}{\text {arcsinh}(a x)}dx}{a}-\frac {x}{a \text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {1}{2} a \left (\frac {\int \frac {\sqrt {a^2 x^2+1}}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}-\frac {x}{a \text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}+\frac {1}{2} a \left (-\frac {x}{a \text {arcsinh}(a x)}+\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {1}{2} a \left (\frac {\text {Chi}(\text {arcsinh}(a x))}{a^2}-\frac {x}{a \text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
-1/2*Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x]^2) + (a*(-(x/(a*ArcSinh[a*x])) + Co shIntegral[ArcSinh[a*x]]/a^2))/2
3.1.64.3.1 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_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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 = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{2 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{2}}{a}\) | \(42\) |
default | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{2 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{2}}{a}\) | \(42\) |
\[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
\[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=\int \frac {1}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
-1/2*(a^7*x^7 + 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 + a^2*x^2)*(a^2*x^2 + 1)^ (3/2) + (3*a^5*x^5 + 5*a^3*x^3 + 2*a*x)*(a^2*x^2 + 1) + a*x + (a^7*x^7 + 3 *a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - 1)*(a^2*x^2 + 1)^(3/2) + 3*(a^5*x^5 + a^ 3*x^3)*(a^2*x^2 + 1) + a*x + (3*a^6*x^6 + 6*a^4*x^4 + 4*a^2*x^2 + 1)*sqrt( a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^6*x^6 + 7*a^4*x^4 + 5*a^ 2*x^2 + 1)*sqrt(a^2*x^2 + 1))/((a^7*x^6 + 3*a^5*x^4 + (a^2*x^2 + 1)^(3/2)* a^4*x^3 + 3*a^3*x^2 + 3*(a^5*x^4 + a^3*x^2)*(a^2*x^2 + 1) + 3*(a^6*x^5 + 2 *a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqrt(a^2*x^2 + 1))^2) + integrate(1/2*(a^8*x^8 + 4*a^6*x^6 + 6*a^4*x^4 + 4*a^2*x^2 + (a^4*x^4 + 3 )*(a^2*x^2 + 1)^2 + (4*a^5*x^5 + 4*a^3*x^3 + 3*a*x)*(a^2*x^2 + 1)^(3/2) + 3*(2*a^6*x^6 + 4*a^4*x^4 + a^2*x^2 - 1)*(a^2*x^2 + 1) + (4*a^7*x^7 + 12*a^ 5*x^5 + 9*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)/((a^8*x^8 + 4*a^6*x^6 + (a ^2*x^2 + 1)^2*a^4*x^4 + 6*a^4*x^4 + 4*a^2*x^2 + 4*(a^5*x^5 + a^3*x^3)*(a^2 *x^2 + 1)^(3/2) + 6*(a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*(a^2*x^2 + 1) + 4*(a^7 *x^7 + 3*a^5*x^5 + 3*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)*log(a*x + sqrt( a^2*x^2 + 1))), x)
\[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^3} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]