Integrand size = 23, antiderivative size = 111 \[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {1}{2} x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}} \] Output:
-1/4*b*c*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/2*x*(c^2*d*x^2+d)^(1/ 2)*(a+b*arcsinh(c*x))+1/4*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^ 2*x^2+1)^(1/2)
Time = 0.38 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} \left (4 a x \sqrt {d+c^2 d x^2}+\frac {4 a \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{c}+\frac {b \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{c \sqrt {1+c^2 x^2}}\right ) \] Input:
Integrate[Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
(4*a*x*Sqrt[d + c^2*d*x^2] + (4*a*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2 *d*x^2]])/c + (b*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c* x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(c*Sqrt[1 + c^2*x^2]))/8
Time = 0.35 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6200, 15, 6198}
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.
\(\displaystyle \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int xdx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {c^2 x^2+1}}-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}\) |
Input:
Int[Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
-1/4*(b*c*x^2*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (x*Sqrt[d + c^2*d*x ^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2) /(4*b*c*Sqrt[1 + c^2*x^2])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(95)=190\).
Time = 0.81 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.31
method | result | size |
default | \(\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}\right )\) | \(256\) |
parts | \(\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}\right )\) | \(256\) |
Input:
int((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/2*a*x*(c^2*d*x^2+d)^(1/2)+1/2*a*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d) ^(1/2))/(c^2*d)^(1/2)+b*(1/4*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*arc sinh(x*c)^2+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1 /2)+2*x*c+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(x*c))/(c^2*x^2+1)/c+1/16*(d*(c^ 2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^( 1/2))*(1+2*arcsinh(x*c))/(c^2*x^2+1)/c)
\[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a), x)
\[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \] Input:
integrate((c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x)),x)
Output:
Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x)), x)
Exception generated. \[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(1/2)*(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. \[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:
int((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2),x)
Output:
int((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2), x)
\[ \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, \left (\sqrt {c^{2} x^{2}+1}\, a c x +2 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b c +\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \right )}{2 c} \] Input:
int((c^2*d*x^2+d)^(1/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*(sqrt(c**2*x**2 + 1)*a*c*x + 2*int(sqrt(c**2*x**2 + 1)*asinh(c*x) ,x)*b*c + log(sqrt(c**2*x**2 + 1) + c*x)*a))/(2*c)