Integrand size = 19, antiderivative size = 75 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d \sqrt {1+c^2 x^2}}{3 c}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{9 c}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x)) \] Output:
-2/3*b*d*(c^2*x^2+1)^(1/2)/c-1/9*b*d*(c^2*x^2+1)^(3/2)/c+d*x*(a+b*arcsinh( c*x))+1/3*c^2*d*x^3*(a+b*arcsinh(c*x))
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=a d x+\frac {1}{3} a c^2 d x^3-\frac {7 b d \sqrt {1+c^2 x^2}}{9 c}-\frac {1}{9} b c d x^2 \sqrt {1+c^2 x^2}+b d x \text {arcsinh}(c x)+\frac {1}{3} b c^2 d x^3 \text {arcsinh}(c x) \] Input:
Integrate[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
a*d*x + (a*c^2*d*x^3)/3 - (7*b*d*Sqrt[1 + c^2*x^2])/(9*c) - (b*c*d*x^2*Sqr t[1 + c^2*x^2])/9 + b*d*x*ArcSinh[c*x] + (b*c^2*d*x^3*ArcSinh[c*x])/3
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.
\(\displaystyle \int \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6199 |
\(\displaystyle -b c \int \frac {d x \left (c^2 x^2+3\right )}{3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} b c d \int \frac {x \left (c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 353 |
\(\displaystyle -\frac {1}{6} b c d \int \frac {c^2 x^2+3}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {1}{6} b c d \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} c^2 d x^3 (a+b \text {arcsinh}(c x))+d x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\) |
Input:
Int[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/6*(b*c*d*((4*Sqrt[1 + c^2*x^2])/c^2 + (2*(1 + c^2*x^2)^(3/2))/(3*c^2))) + d*x*(a + b*ArcSinh[c*x]) + (c^2*d*x^3*(a + b*ArcSinh[c*x]))/3
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_.)*((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_.))*((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]
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97
method | result | size |
parts | \(a d \left (\frac {1}{3} x^{3} c^{2}+x \right )+\frac {b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(73\) |
derivativedivides | \(\frac {a d \left (\frac {1}{3} x^{3} c^{3}+x c \right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(76\) |
default | \(\frac {a d \left (\frac {1}{3} x^{3} c^{3}+x c \right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c}\) | \(76\) |
orering | \(\frac {x \left (5 c^{2} x^{2}+23\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9 c^{2} x^{2}+9}-\frac {\left (c^{2} x^{2}+7\right ) \left (2 c^{2} d x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {\left (c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{9 c^{2}}\) | \(98\) |
Input:
int((c^2*d*x^2+d)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
a*d*(1/3*x^3*c^2+x)+b*d/c*(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 = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, a c^{3} d x^{3} + 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} + 3 \, b c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{2} d x^{2} + 7 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, c} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/9*(3*a*c^3*d*x^3 + 9*a*c*d*x + 3*(b*c^3*d*x^3 + 3*b*c*d*x)*log(c*x + sqr t(c^2*x^2 + 1)) - (b*c^2*d*x^2 + 7*b*d)*sqrt(c^2*x^2 + 1))/c
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{3}}{3} + a d x + \frac {b c^{2} d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b c d x^{2} \sqrt {c^{2} x^{2} + 1}}{9} + b d x \operatorname {asinh}{\left (c x \right )} - \frac {7 b d \sqrt {c^{2} x^{2} + 1}}{9 c} & \text {for}\: c \neq 0 \\a d x & \text {otherwise} \end {cases} \] Input:
integrate((c**2*d*x**2+d)*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*c**2*d*x**3/3 + a*d*x + b*c**2*d*x**3*asinh(c*x)/3 - b*c*d*x* *2*sqrt(c**2*x**2 + 1)/9 + b*d*x*asinh(c*x) - 7*b*d*sqrt(c**2*x**2 + 1)/(9 *c), Ne(c, 0)), (a*d*x, True))
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a c^{2} d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/3*a*c^2*d*x^3 + 1/9*(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))*b*c^2*d + a*d*x + (c*x*arcsinh(c*x) - sqrt(c^2* x^2 + 1))*b*d/c
Exception generated. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x)),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. \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \] Input:
int((a + b*asinh(c*x))*(d + c^2*d*x^2),x)
Output:
int((a + b*asinh(c*x))*(d + c^2*d*x^2), x)
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (3 \mathit {asinh} \left (c x \right ) b \,c^{3} x^{3}+9 \mathit {asinh} \left (c x \right ) b c x -\sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, b +3 a \,c^{3} x^{3}+9 a c x \right )}{9 c} \] Input:
int((c^2*d*x^2+d)*(a+b*asinh(c*x)),x)
Output:
(d*(3*asinh(c*x)*b*c**3*x**3 + 9*asinh(c*x)*b*c*x - sqrt(c**2*x**2 + 1)*b* c**2*x**2 - 7*sqrt(c**2*x**2 + 1)*b + 3*a*c**3*x**3 + 9*a*c*x))/(9*c)