Integrand size = 21, antiderivative size = 125 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{9 c}+\frac {2}{3} d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \]
14/9*b^2*d*x+2/27*b^2*c^2*d*x^3-2/9*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 -4/3*b*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (9 a^2 c x \left (3+c^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )+2 b^2 c x \left (21+c^2 x^2\right )-6 b \left (-3 a c x \left (3+c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )\right ) \text {arcsinh}(c x)+9 b^2 c x \left (3+c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{27 c} \]
(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 = 0.57 (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 |
\(\displaystyle \frac {1}{3} d x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c d \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\) |
(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
3.3.2.3.1 Defintions of rubi rules used
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 = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}+c x \right )+d \,b^{2} \left (\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 c x}{27}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c^{2} x^{2}+1\right )}{27}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
default | \(\frac {d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}+c x \right )+d \,b^{2} \left (\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 c x}{27}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c^{2} x^{2}+1\right )}{27}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
parts | \(d \,a^{2} \left (\frac {1}{3} x^{3} c^{2}+x \right )+\frac {d \,b^{2} \left (\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{3}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 c x}{27}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c^{2} x^{2}+1\right )}{27}\right )}{c}+\frac {2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(166\) |
1/c*(d*a^2*(1/3*c^3*x^3+c*x)+d*b^2*(2/3*arcsinh(c*x)^2*x*c+1/3*arcsinh(c*x )^2*c*x*(c^2*x^2+1)-4/3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+40/27*c*x-2/9*arcsi nh(c*x)*(c^2*x^2+1)^(3/2)+2/27*c*x*(c^2*x^2+1))+2*d*a*b*(1/3*arcsinh(c*x)* c^3*x^3+arcsinh(c*x)*c*x-1/9*c^2*x^2*(c^2*x^2+1)^(1/2)-7/9*(c^2*x^2+1)^(1/ 2)))
Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} d x^{3} + 3 \, {\left (9 \, a^{2} + 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} + 3 \, b^{2} c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{3} d x^{3} + 9 \, a b c d x - {\left (b^{2} c^{2} d x^{2} + 7 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (a b c^{2} d x^{2} + 7 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c} \]
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 = 0.23 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.79 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{3}}{3} + a^{2} d x + \frac {2 a b c^{2} d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b c d x^{2} \sqrt {c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname {asinh}{\left (c x \right )} - \frac {14 a b d \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b^{2} c^{2} d x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} c^{2} d x^{3}}{27} - \frac {2 b^{2} c d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {14 b^{2} d x}{9} - \frac {14 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} & \text {for}\: c \neq 0 \\a^{2} d x & \text {otherwise} \end {cases} \]
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 = 0.21 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.84 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} c^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} c^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \]
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. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \]