Integrand size = 23, antiderivative size = 214 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \]
298/225*b^2*d^2*x+76/675*b^2*c^2*d^2*x^3+2/125*b^2*c^4*d^2*x^5-8/45*b*d^2* (c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c-2/25*b*d^2*(c^2*x^2+1)^(5/2)*(a+b*a rcsinh(c*x))/c+8/15*d^2*x*(a+b*arcsinh(c*x))^2+4/15*d^2*x*(c^2*x^2+1)*(a+b *arcsinh(c*x))^2+1/5*d^2*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2-16/15*b*d^2* (a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 0.93 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.89 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (225 a^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235+190 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{3375 c} \]
(d^2*(225*a^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 + c^2*x^2] *(149 + 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(2235 + 190*c^2*x^2 + 27*c^4*x ^4) - 30*b*(-15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 + c^2*x^2]* (149 + 38*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c*x*(15 + 10*c^2*x^ 2 + 3*c^4*x^4)*ArcSinh[c*x]^2))/(3375*c)
Time = 0.92 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6201, 27, 6201, 6187, 6213, 24, 210, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{5} b c d^2 \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} d \int d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} b c d^2 \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} d^2 \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{5} b c d^2 \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} d^2 \left (-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} \int (a+b \text {arcsinh}(c x))^2dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c d^2 \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {4}{5} d^2 \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {4}{5} d^2 \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^4 x^4+2 c^2 x^2+1\right )dx}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\) |
(d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*d^2*(-1/5*(b*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/c + ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c* x]))/(5*c^2)))/5 + (4*d^2*((x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (2 *b*c*(-1/3*(b*(x + (c^2*x^3)/3))/c + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c *x]))/(3*c^2)))/3 + (2*(x*(a + b*ArcSinh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sq rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2)))/3))/5
3.3.11.3.1 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_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 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_.))^(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 = 0.20 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
| parts | \(d^{2} a^{2} \left (\frac {1}{5} c^{4} x^{5}+\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
1/c*(d^2*a^2*(1/5*c^5*x^5+2/3*c^3*x^3+c*x)+d^2*b^2*(8/15*arcsinh(c*x)^2*x* c+1/5*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2+4/15*arcsinh(c*x)^2*c*x*(c^2*x^2+1) -16/15*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+4144/3375*c*x-2/25*arcsinh(c*x)*(c^2 *x^2+1)^(5/2)+2/125*c*x*(c^2*x^2+1)^2+272/3375*c*x*(c^2*x^2+1)-8/45*arcsin h(c*x)*(c^2*x^2+1)^(3/2))+2*d^2*a*b*(1/5*arcsinh(c*x)*c^5*x^5+2/3*arcsinh( c*x)*c^3*x^3+arcsinh(c*x)*c*x-149/225*(c^2*x^2+1)^(1/2)-1/25*c^4*x^4*(c^2* x^2+1)^(1/2)-38/225*c^2*x^2*(c^2*x^2+1)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.30 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x^{5} + 10 \, {\left (225 \, a^{2} + 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} + 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} + 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d^{2} x^{5} + 150 \, a b c^{3} d^{2} x^{3} + 225 \, a b c d^{2} x - {\left (9 \, b^{2} c^{4} d^{2} x^{4} + 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} + 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c} \]
1/3375*(27*(25*a^2 + 2*b^2)*c^5*d^2*x^5 + 10*(225*a^2 + 38*b^2)*c^3*d^2*x^ 3 + 15*(225*a^2 + 298*b^2)*c*d^2*x + 225*(3*b^2*c^5*d^2*x^5 + 10*b^2*c^3*d ^2*x^3 + 15*b^2*c*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^5*d ^2*x^5 + 150*a*b*c^3*d^2*x^3 + 225*a*b*c*d^2*x - (9*b^2*c^4*d^2*x^4 + 38*b ^2*c^2*d^2*x^2 + 149*b^2*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*d^2*x^4 + 38*a*b*c^2*d^2*x^2 + 149*a*b*d^2)*sqrt(c^2*x ^2 + 1))/c
Time = 0.46 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.82 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{5}}{5} + \frac {2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac {2 a b c^{4} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {4 a b c^{2} d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {76 a b c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} - \frac {298 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {b^{2} c^{4} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{4} d^{2} x^{5}}{125} - \frac {2 b^{2} c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25} + \frac {2 b^{2} c^{2} d^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac {76 b^{2} c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {298 b^{2} d^{2} x}{225} - \frac {298 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c} & \text {for}\: c \neq 0 \\a^{2} d^{2} x & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**4*d**2*x**5/5 + 2*a**2*c**2*d**2*x**3/3 + a**2*d**2*x + 2*a*b*c**4*d**2*x**5*asinh(c*x)/5 - 2*a*b*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)/25 + 4*a*b*c**2*d**2*x**3*asinh(c*x)/3 - 76*a*b*c*d**2*x**2*sqrt(c**2* x**2 + 1)/225 + 2*a*b*d**2*x*asinh(c*x) - 298*a*b*d**2*sqrt(c**2*x**2 + 1) /(225*c) + b**2*c**4*d**2*x**5*asinh(c*x)**2/5 + 2*b**2*c**4*d**2*x**5/125 - 2*b**2*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/25 + 2*b**2*c**2*d **2*x**3*asinh(c*x)**2/3 + 76*b**2*c**2*d**2*x**3/675 - 76*b**2*c*d**2*x** 2*sqrt(c**2*x**2 + 1)*asinh(c*x)/225 + b**2*d**2*x*asinh(c*x)**2 + 298*b** 2*d**2*x/225 - 298*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(225*c), Ne(c, 0)), (a**2*d**2*x, True))
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (190) = 380\).
Time = 0.23 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.14 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} + \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} + \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
1/5*b^2*c^4*d^2*x^5*arcsinh(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 + 2/3*b^2*c^2*d^2 *x^3*arcsinh(c*x)^2 + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4 /c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^4*d ^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120 *x)/c^4)*b^2*c^4*d^2 + 2/3*a^2*c^2*d^2*x^3 + 4/9*(3*x^3*arcsinh(c*x) - c*( sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^2*d^2 - 4/27*( 3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - ( c^2*x^3 - 6*x)/c^2)*b^2*c^2*d^2 + b^2*d^2*x*arcsinh(c*x)^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsinh(c*x) - sq rt(c^2*x^2 + 1))*a*b*d^2/c
Exception generated. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]