Integrand size = 21, antiderivative size = 125 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{9 c}+\frac {2}{3} d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \] Output:
14/9*b^2*d*x+2/27*b^2*c^2*d*x^3-4/3*b*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x ))/c-2/9*b*d*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c+2/3*d*x*(a+b*arcsinh(c *x))^2+1/3*d*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (9 a^2 c x \left (3+c^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )+2 b^2 c x \left (21+c^2 x^2\right )-6 b \left (-3 a c x \left (3+c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )\right ) \text {arcsinh}(c x)+9 b^2 c x \left (3+c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{27 c} \] Input:
Integrate[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d*(9*a^2*c*x*(3 + c^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(7 + c^2*x^2) + 2*b^ 2*c*x*(21 + c^2*x^2) - 6*b*(-3*a*c*x*(3 + c^2*x^2) + b*Sqrt[1 + c^2*x^2]*( 7 + c^2*x^2))*ArcSinh[c*x] + 9*b^2*c*x*(3 + c^2*x^2)*ArcSinh[c*x]^2))/(27* c)
Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6201, 6187, 6213, 24, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{3} b c d \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} d \int (a+b \text {arcsinh}(c x))^2dx+\frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {2}{3} d \left (x (a+b \text {arcsinh}(c x))^2-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{3} b c d \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {2}{3} d \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )-\frac {2}{3} b c d \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {2}{3} b c d \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c d \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\) |
Input:
Int[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*d*(-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* d*(x*(a + b*ArcSinh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2)))/3
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 = 1.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {a^{2} d \left (\frac {1}{3} x^{3} c^{3}+x c \right )+b^{2} d \left (\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 x c}{27}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 x c \left (c^{2} x^{2}+1\right )}{27}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
| default | \(\frac {a^{2} d \left (\frac {1}{3} x^{3} c^{3}+x c \right )+b^{2} d \left (\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 x c}{27}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 x c \left (c^{2} x^{2}+1\right )}{27}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
| parts | \(a^{2} d \left (\frac {1}{3} x^{3} c^{2}+x \right )+\frac {b^{2} d \left (\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 x c}{27}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 x c \left (c^{2} x^{2}+1\right )}{27}\right )}{c}+\frac {2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
| orering | \(\frac {x \left (19 c^{4} x^{4}+166 c^{2} x^{2}+27\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{27 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (2 c^{4} x^{4}+29 c^{2} x^{2}+7\right ) \left (2 c^{2} d x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {2 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{9 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {x \left (c^{2} x^{2}+21\right ) \left (2 c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {8 c^{3} d x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \left (c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3} x}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{2}}\) | \(263\) |
Input:
int((c^2*d*x^2+d)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(a^2*d*(1/3*x^3*c^3+x*c)+b^2*d*(2/3*arcsinh(x*c)^2*x*c+1/3*arcsinh(x*c )^2*x*c*(c^2*x^2+1)-4/3*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+40/27*x*c-2/9*arcsi nh(x*c)*(c^2*x^2+1)^(3/2)+2/27*x*c*(c^2*x^2+1))+2*a*b*d*(1/3*arcsinh(x*c)* x^3*c^3+x*c*arcsinh(x*c)-1/9*x^2*c^2*(c^2*x^2+1)^(1/2)-7/9*(c^2*x^2+1)^(1/ 2)))
Time = 0.11 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} d x^{3} + 3 \, {\left (9 \, a^{2} + 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} + 3 \, b^{2} c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{3} d x^{3} + 9 \, a b c d x - {\left (b^{2} c^{2} d x^{2} + 7 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (a b c^{2} d x^{2} + 7 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/27*((9*a^2 + 2*b^2)*c^3*d*x^3 + 3*(9*a^2 + 14*b^2)*c*d*x + 9*(b^2*c^3*d* x^3 + 3*b^2*c*d*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(3*a*b*c^3*d*x^3 + 9 *a*b*c*d*x - (b^2*c^2*d*x^2 + 7*b^2*d)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c ^2*x^2 + 1)) - 6*(a*b*c^2*d*x^2 + 7*a*b*d)*sqrt(c^2*x^2 + 1))/c
Time = 0.28 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.79 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{3}}{3} + a^{2} d x + \frac {2 a b c^{2} d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b c d x^{2} \sqrt {c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname {asinh}{\left (c x \right )} - \frac {14 a b d \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b^{2} c^{2} d x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} c^{2} d x^{3}}{27} - \frac {2 b^{2} c d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {14 b^{2} d x}{9} - \frac {14 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} & \text {for}\: c \neq 0 \\a^{2} d x & \text {otherwise} \end {cases} \] Input:
integrate((c**2*d*x**2+d)*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**2*d*x**3/3 + a**2*d*x + 2*a*b*c**2*d*x**3*asinh(c*x)/3 - 2*a*b*c*d*x**2*sqrt(c**2*x**2 + 1)/9 + 2*a*b*d*x*asinh(c*x) - 14*a*b*d*s qrt(c**2*x**2 + 1)/(9*c) + b**2*c**2*d*x**3*asinh(c*x)**2/3 + 2*b**2*c**2* d*x**3/27 - 2*b**2*c*d*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/9 + b**2*d*x*as inh(c*x)**2 + 14*b**2*d*x/9 - 14*b**2*d*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9* c), Ne(c, 0)), (a**2*d*x, True))
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (109) = 218\).
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.84 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} c^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} c^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/3*b^2*c^2*d*x^3*arcsinh(c*x)^2 + 1/3*a^2*c^2*d*x^3 + 2/9*(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 - 2/27*(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 + b^2*d*x*arcsinh(c*x)^2 + 2*b^2*d*( x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d*x + 2*(c*x*arcsinh(c*x) - sq rt(c^2*x^2 + 1))*a*b*d/c
Exception generated. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
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. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \] Input:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2),x)
Output:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2), x)
\[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (9 \mathit {asinh} \left (c x \right )^{2} b^{2} c x -18 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2}+6 \mathit {asinh} \left (c x \right ) a b \,c^{3} x^{3}+18 \mathit {asinh} \left (c x \right ) a b c x -2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-14 \sqrt {c^{2} x^{2}+1}\, a b +9 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+3 a^{2} c^{3} x^{3}+9 a^{2} c x +18 b^{2} c x \right )}{9 c} \] Input:
int((c^2*d*x^2+d)*(a+b*asinh(c*x))^2,x)
Output:
(d*(9*asinh(c*x)**2*b**2*c*x - 18*sqrt(c**2*x**2 + 1)*asinh(c*x)*b**2 + 6* asinh(c*x)*a*b*c**3*x**3 + 18*asinh(c*x)*a*b*c*x - 2*sqrt(c**2*x**2 + 1)*a *b*c**2*x**2 - 14*sqrt(c**2*x**2 + 1)*a*b + 9*int(asinh(c*x)**2*x**2,x)*b* *2*c**3 + 3*a**2*c**3*x**3 + 9*a**2*c*x + 18*b**2*c*x))/(9*c)