\(\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx\) [96]

Optimal result

Integrand size = 26, antiderivative size = 56 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \] Output:

-2*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/Pi^(1/2)-b*polylog(2, 
-c*x-(c^2*x^2+1)^(1/2))/Pi^(1/2)+b*polylog(2,c*x+(c^2*x^2+1)^(1/2))/Pi^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.71 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {b \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-b \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-b \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \] Input:

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

Output:

(b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - b*ArcSinh[c*x]*Log[1 + E^(-Ar 
cSinh[c*x])] + a*Log[x] - a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*PolyLog[2, 
 -E^(-ArcSinh[c*x])] - b*PolyLog[2, E^(-ArcSinh[c*x])])/Sqrt[Pi]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6231, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6231

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)}{\sqrt {\pi }}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {\pi }}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {\pi }}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{\sqrt {\pi }}\)

Input:

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

Output:

(I*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E 
^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]))/Sqrt[Pi]
 

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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6231

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.11

method	result	size
default	\(-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {\pi }}\)	\(118\)
parts	\(-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {\pi }}\)	\(118\)

Input:

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

Output:

-a/Pi^(1/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))+b/Pi^(1/2)*(arcsinh(x* 
c)*ln(1-x*c-(c^2*x^2+1)^(1/2))+polylog(2,x*c+(c^2*x^2+1)^(1/2))-arcsinh(x* 
c)*ln(1+x*c+(c^2*x^2+1)^(1/2))-polylog(2,-x*c-(c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi*c^2*x^3 + pi*x), x 
)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*x),x)*b + log(sqrt(c**2*x**2 + 1) + c 
*x - 1)*a - log(sqrt(c**2*x**2 + 1) + c*x + 1)*a)/sqrt(pi)