\(\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx\) [86]

Optimal result

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

[Out]

-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*p
olylog(2,c*x+(c^2*x^2+1)^(1/2))/Pi^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5816, 4267, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {\pi }}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 5816

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{\sqrt {\pi }} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {\pi }} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \\ & = -\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 }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.71 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {b \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-b \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-b \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

(b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - b*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[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]

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.11

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

(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]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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