\(\int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [424]

Optimal result

Integrand size = 25, antiderivative size = 73 \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2} \] Output:

-x/b/c/(a+b*arcsinh(c*x))+cosh(a/b)*Chi((a+b*arcsinh(c*x))/b)/b^2/c^2-sinh 
(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {b c x}{a+b \text {arcsinh}(c x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b^2 c^2} \] Input:

Integrate[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]
 

Output:

(-((b*c*x)/(a + b*ArcSinh[c*x])) + Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c* 
x]] - Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c^2)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6233, 6189, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

\(\displaystyle \int \frac {x}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6233

\(\displaystyle \frac {\int \frac {1}{a+b \text {arcsinh}(c x)}dx}{b c}-\frac {x}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6189

\(\displaystyle \frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}-\frac {x}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}-\frac {x}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {x}{b c (a+b \text {arcsinh}(c x))}\)

Input:

Int[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]
 

Output:

-(x/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c 
*x])/b] - Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c^2)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 6189

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6233

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c 
^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* 
(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[(f*x)^(m - 1)*(a + 
b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e 
, c^2*d] && LtQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(73)=146\).

Time = 1.47 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.07

method	result	size
default	\(-\frac {x c -\sqrt {c^{2} x^{2}+1}}{2 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 c^{2} b^{2}}-\frac {\operatorname {arcsinh}\left (x c \right ) {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) b +{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) a +b x c +\sqrt {c^{2} x^{2}+1}\, b}{2 c^{2} b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}\)	\(151\)

Input:

int(x/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/2*(x*c-(c^2*x^2+1)^(1/2))/c^2/b/(a+b*arcsinh(x*c))-1/2/c^2/b^2*exp(a/b) 
*Ei(1,arcsinh(x*c)+a/b)-1/2/c^2/b^2*(arcsinh(x*c)*exp(-a/b)*Ei(1,-arcsinh( 
x*c)-a/b)*b+exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*a+b*x*c+(c^2*x^2+1)^(1/2)*b) 
/(a+b*arcsinh(x*c))
 

Fricas [F]

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

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

Output:

integral(sqrt(c^2*x^2 + 1)*x/(a^2*c^2*x^2 + (b^2*c^2*x^2 + b^2)*arcsinh(c* 
x)^2 + a^2 + 2*(a*b*c^2*x^2 + a*b)*arcsinh(c*x)), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*x^4 + c*x^2 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)*a*b*c^ 
2*x + ((c^2*x^2 + 1)*b^2*c^2*x + (b^2*c^3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1))* 
log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)) + 
integrate((c^5*x^5 + (c^2*x^2 + 1)*c^3*x^3 + 3*c^3*x^3 + 2*c*x + (2*c^4*x^ 
4 + 3*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^2 + 2 
*(a*b*c^4*x^3 + a*b*c^2*x)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^3*x^ 
2 + 2*(b^2*c^4*x^3 + b^2*c^2*x)*(c^2*x^2 + 1) + (b^2*c^5*x^4 + 2*b^2*c^3*x 
^2 + b^2*c)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^4 
 + 2*a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)), x)
 

Giac [F]

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

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

Output:

integrate(x/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)^2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(x/(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*b**2 + 2*sqrt(c**2*x**2 + 1)*asin 
h(c*x)*a*b + sqrt(c**2*x**2 + 1)*a**2),x)