Integrand size = 21, antiderivative size = 205 \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=-\frac {3 a x^2 \sqrt {c+a^2 c x^2}}{8 \sqrt {1+a^2 x^2}}+\frac {3}{4} x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {3 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{8 a \sqrt {1+a^2 x^2}}-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{4 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{8 a \sqrt {1+a^2 x^2}} \]
3/4*x*arcsinh(a*x)*(a^2*c*x^2+c)^(1/2)+1/2*x*arcsinh(a*x)^3*(a^2*c*x^2+c)^ (1/2)-3/8*a*x^2*(a^2*c*x^2+c)^(1/2)/(a^2*x^2+1)^(1/2)-3/8*arcsinh(a*x)^2*( a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)-3/4*a*x^2*arcsinh(a*x)^2*(a^2*c*x^2 +c)^(1/2)/(a^2*x^2+1)^(1/2)+1/8*arcsinh(a*x)^4*(a^2*c*x^2+c)^(1/2)/a/(a^2* x^2+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.42 \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-3 \left (1+2 \text {arcsinh}(a x)^2\right ) \cosh (2 \text {arcsinh}(a x))+2 \text {arcsinh}(a x) \left (\text {arcsinh}(a x)^3+\left (3+2 \text {arcsinh}(a x)^2\right ) \sinh (2 \text {arcsinh}(a x))\right )\right )}{16 a \sqrt {1+a^2 x^2}} \]
(Sqrt[c*(1 + a^2*x^2)]*(-3*(1 + 2*ArcSinh[a*x]^2)*Cosh[2*ArcSinh[a*x]] + 2 *ArcSinh[a*x]*(ArcSinh[a*x]^3 + (3 + 2*ArcSinh[a*x]^2)*Sinh[2*ArcSinh[a*x] ])))/(16*a*Sqrt[1 + a^2*x^2])
Time = 0.81 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6200, 6191, 6198, 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.
\(\displaystyle \int \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c} \, dx\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \int x \text {arcsinh}(a x)^2dx}{2 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )\right )}{2 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {\text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 a \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right ) \sqrt {a^2 c x^2+c}}{2 \sqrt {a^2 x^2+1}}\) |
(x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a* x]^4)/(8*a*Sqrt[1 + a^2*x^2]) - (3*a*Sqrt[c + a^2*c*x^2]*((x^2*ArcSinh[a*x ]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcS inh[a*x]^2/(4*a^3))))/(2*Sqrt[1 + a^2*x^2])
3.4.36.3.1 Defintions of rubi rules used
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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
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 = 0.22 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4}}{8 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}-6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-3\right )}{32 \left (a^{2} x^{2}+1\right ) a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+3\right )}{32 \left (a^{2} x^{2}+1\right ) a}\) | \(231\) |
1/8*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a*arcsinh(a*x)^4+1/32*(c*(a^2* x^2+1))^(1/2)*(2*a^3*x^3+2*a^2*x^2*(a^2*x^2+1)^(1/2)+2*a*x+(a^2*x^2+1)^(1/ 2))*(4*arcsinh(a*x)^3-6*arcsinh(a*x)^2+6*arcsinh(a*x)-3)/(a^2*x^2+1)/a+1/3 2*(c*(a^2*x^2+1))^(1/2)*(2*a^3*x^3-2*a^2*x^2*(a^2*x^2+1)^(1/2)+2*a*x-(a^2* x^2+1)^(1/2))*(4*arcsinh(a*x)^3+6*arcsinh(a*x)^2+6*arcsinh(a*x)+3)/(a^2*x^ 2+1)/a
\[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
\[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]