Integrand size = 17, antiderivative size = 153 \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=-\frac {40 c \sqrt {1+a^2 x^2}}{9 a}-\frac {2 c \left (1+a^2 x^2\right )^{3/2}}{27 a}+\frac {14}{3} c x \text {arcsinh}(a x)+\frac {2}{9} a^2 c x^3 \text {arcsinh}(a x)-\frac {2 c \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a}-\frac {c \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arcsinh}(a x)^3+\frac {1}{3} c x \left (1+a^2 x^2\right ) \text {arcsinh}(a x)^3 \]
-2/27*c*(a^2*x^2+1)^(3/2)/a+14/3*c*x*arcsinh(a*x)+2/9*a^2*c*x^3*arcsinh(a* x)-1/3*c*(a^2*x^2+1)^(3/2)*arcsinh(a*x)^2/a+2/3*c*x*arcsinh(a*x)^3+1/3*c*x *(a^2*x^2+1)*arcsinh(a*x)^3-40/9*c*(a^2*x^2+1)^(1/2)/a-2*c*arcsinh(a*x)^2* (a^2*x^2+1)^(1/2)/a
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=\frac {c \left (-2 \sqrt {1+a^2 x^2} \left (61+a^2 x^2\right )+6 a x \left (21+a^2 x^2\right ) \text {arcsinh}(a x)-9 \sqrt {1+a^2 x^2} \left (7+a^2 x^2\right ) \text {arcsinh}(a x)^2+9 a x \left (3+a^2 x^2\right ) \text {arcsinh}(a x)^3\right )}{27 a} \]
(c*(-2*Sqrt[1 + a^2*x^2]*(61 + a^2*x^2) + 6*a*x*(21 + a^2*x^2)*ArcSinh[a*x ] - 9*Sqrt[1 + a^2*x^2]*(7 + a^2*x^2)*ArcSinh[a*x]^2 + 9*a*x*(3 + a^2*x^2) *ArcSinh[a*x]^3))/(27*a)
Time = 0.82 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {6201, 6187, 6213, 6187, 241, 6199, 27, 353, 53, 2009}
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.
| \(\displaystyle \int \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right ) \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -a c \int x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2dx+\frac {2}{3} c \int \text {arcsinh}(a x)^3dx+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx\right )-a c \int x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2dx+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \int \text {arcsinh}(a x)dx}{a}\right )\right )-a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-a \int \frac {x}{\sqrt {a^2 x^2+1}}dx\right )}{a}\right )\right )-a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 6199 |
| \(\displaystyle -a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-a \int \frac {x \left (a^2 x^2+3\right )}{3 \sqrt {a^2 x^2+1}}dx+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a \int \frac {x \left (a^2 x^2+3\right )}{\sqrt {a^2 x^2+1}}dx+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle -a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \frac {a^2 x^2+3}{\sqrt {a^2 x^2+1}}dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {a^2 x^2+1}+\frac {2}{\sqrt {a^2 x^2+1}}\right )dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} c \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )-a c \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^2}+\frac {4 \sqrt {a^2 x^2+1}}{a^2}\right )+x \text {arcsinh}(a x)\right )}{3 a}\right )\) |
(c*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/3 - a*c*(((1 + a^2*x^2)^(3/2)*ArcSinh[a *x]^2)/(3*a^2) - (2*(-1/6*(a*((4*Sqrt[1 + a^2*x^2])/a^2 + (2*(1 + a^2*x^2) ^(3/2))/(3*a^2))) + x*ArcSinh[a*x] + (a^2*x^3*ArcSinh[a*x])/3))/(3*a)) + ( 2*c*(x*ArcSinh[a*x]^3 - 3*a*((Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a^2 - (2*( -(Sqrt[1 + a^2*x^2]/a) + x*ArcSinh[a*x]))/a)))/3
3.4.30.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
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_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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 = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {c \left (9 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}-9 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+27 a x \operatorname {arcsinh}\left (a x \right )^{3}+6 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )-63 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+126 a x \,\operatorname {arcsinh}\left (a x \right )-122 \sqrt {a^{2} x^{2}+1}\right )}{27 a}\) | \(128\) |
| default | \(\frac {c \left (9 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}-9 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+27 a x \operatorname {arcsinh}\left (a x \right )^{3}+6 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )-63 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+126 a x \,\operatorname {arcsinh}\left (a x \right )-122 \sqrt {a^{2} x^{2}+1}\right )}{27 a}\) | \(128\) |
1/27/a*c*(9*a^3*x^3*arcsinh(a*x)^3-9*a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1 /2)+27*a*x*arcsinh(a*x)^3+6*a^3*x^3*arcsinh(a*x)-63*arcsinh(a*x)^2*(a^2*x^ 2+1)^(1/2)-2*a^2*x^2*(a^2*x^2+1)^(1/2)+126*a*x*arcsinh(a*x)-122*(a^2*x^2+1 )^(1/2))
Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=\frac {9 \, {\left (a^{3} c x^{3} + 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 9 \, {\left (a^{2} c x^{2} + 7 \, c\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (a^{3} c x^{3} + 21 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (a^{2} c x^{2} + 61 \, c\right )} \sqrt {a^{2} x^{2} + 1}}{27 \, a} \]
1/27*(9*(a^3*c*x^3 + 3*a*c*x)*log(a*x + sqrt(a^2*x^2 + 1))^3 - 9*(a^2*c*x^ 2 + 7*c)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2 + 6*(a^3*c*x^3 + 21*a*c*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 2*(a^2*c*x^2 + 61*c)*sqrt(a^2*x^ 2 + 1))/a
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98 \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {a^{2} c x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {asinh}{\left (a x \right )}}{9} - \frac {a c x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3} - \frac {2 a c x^{2} \sqrt {a^{2} x^{2} + 1}}{27} + c x \operatorname {asinh}^{3}{\left (a x \right )} + \frac {14 c x \operatorname {asinh}{\left (a x \right )}}{3} - \frac {7 c \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**2*c*x**3*asinh(a*x)**3/3 + 2*a**2*c*x**3*asinh(a*x)/9 - a*c* x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/3 - 2*a*c*x**2*sqrt(a**2*x**2 + 1)/ 27 + c*x*asinh(a*x)**3 + 14*c*x*asinh(a*x)/3 - 7*c*sqrt(a**2*x**2 + 1)*asi nh(a*x)**2/(3*a) - 122*c*sqrt(a**2*x**2 + 1)/(27*a), Ne(a, 0)), (0, True))
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.81 \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=-\frac {1}{3} \, {\left (\sqrt {a^{2} x^{2} + 1} c x^{2} + \frac {7 \, \sqrt {a^{2} x^{2} + 1} c}{a^{2}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{3} \, {\left (a^{2} c x^{3} + 3 \, c x\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {2}{27} \, {\left (\sqrt {a^{2} x^{2} + 1} c x^{2} - \frac {3 \, {\left (a^{2} c x^{3} + 21 \, c x\right )} \operatorname {arsinh}\left (a x\right )}{a} + \frac {61 \, \sqrt {a^{2} x^{2} + 1} c}{a^{2}}\right )} a \]
-1/3*(sqrt(a^2*x^2 + 1)*c*x^2 + 7*sqrt(a^2*x^2 + 1)*c/a^2)*a*arcsinh(a*x)^ 2 + 1/3*(a^2*c*x^3 + 3*c*x)*arcsinh(a*x)^3 - 2/27*(sqrt(a^2*x^2 + 1)*c*x^2 - 3*(a^2*c*x^3 + 21*c*x)*arcsinh(a*x)/a + 61*sqrt(a^2*x^2 + 1)*c/a^2)*a
Exception generated. \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (c+a^2 c x^2\right ) \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]