\(\int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) [92]

Optimal result

Integrand size = 23, antiderivative size = 198 \[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c}+\frac {d e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c}+\frac {d e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c} \] Output:

3/8*d*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c+1/ 
24*d*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1 
/2))/b^(1/2)/c+3/8*d*Pi^(1/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(1/ 
2)/c/exp(a/b)+1/24*d*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2 
)/b^(1/2))/b^(1/2)/c/exp(3*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99 \[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {d e^{-\frac {3 a}{b}} \left (-9 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{24 c \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[(d + c^2*d*x^2)/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

(d*(-9*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] 
 + Sqrt[3]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c 
*x]))/b] + 9*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + 
 b*ArcSinh[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[ 
1/2, (3*(a + b*ArcSinh[c*x]))/b]))/(24*c*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c* 
x]])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6206, 3042, 3793, 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.

\(\displaystyle \int \frac {c^2 d x^2+d}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\)

\(\Big \downarrow \) 6206

\(\displaystyle \frac {d \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {d \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \left (\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b c}\)

Input:

Int[(d + c^2*d*x^2)/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

(d*((3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/8 + 
 (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sq 
rt[b]])/8 + (3*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8 
*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sq 
rt[b]])/(8*E^((3*a)/b))))/(b*c)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
 

Maple [F]

\[\int \frac {c^{2} d \,x^{2}+d}{\sqrt {a +b \,\operatorname {arcsinh}\left (x c \right )}}d x\]

Input:

int((c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(1/2),x)
 

Output:

int((c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(1/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=d \left (\int \frac {c^{2} x^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx + \int \frac {1}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx\right ) \] Input:

integrate((c**2*d*x**2+d)/(a+b*asinh(c*x))**(1/2),x)
 

Output:

d*(Integral(c**2*x**2/sqrt(a + b*asinh(c*x)), x) + Integral(1/sqrt(a + b*a 
sinh(c*x)), x))
 

Maxima [F]

\[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {c^{2} d x^{2} + d}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \] Input:

integrate((c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")
 

Output:

integrate((c^2*d*x^2 + d)/sqrt(b*arcsinh(c*x) + a), x)
 

Giac [F]

\[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {c^{2} d x^{2} + d}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \] Input:

integrate((c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")
 

Output:

integrate((c^2*d*x^2 + d)/sqrt(b*arcsinh(c*x) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {d\,c^2\,x^2+d}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \] Input:

int((d + c^2*d*x^2)/(a + b*asinh(c*x))^(1/2),x)
 

Output:

int((d + c^2*d*x^2)/(a + b*asinh(c*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {d+c^2 d x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=d \left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}}{\mathit {asinh} \left (c x \right ) b +a}d x +\left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{2}}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) c^{2}\right ) \] Input:

int((c^2*d*x^2+d)/(a+b*asinh(c*x))^(1/2),x)
 

Output:

d*(int(sqrt(asinh(c*x)*b + a)/(asinh(c*x)*b + a),x) + int((sqrt(asinh(c*x) 
*b + a)*x**2)/(asinh(c*x)*b + a),x)*c**2)