\(\int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx\) [59]

Optimal result

Integrand size = 26, antiderivative size = 89 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=-b c \sqrt {\pi } x+\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-2 \sqrt {\pi } (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-b \sqrt {\pi } \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+b \sqrt {\pi } \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5806, 5816, 4267, 2317, 2438, 8} \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=-2 \sqrt {\pi } \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\sqrt {\pi } b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\sqrt {\pi } b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )+\sqrt {\pi } (-b) c x \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 5806

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]
/Sqrt[1 + c^2*x^2]], Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))
*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*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] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\sqrt {\pi } \left (a \sqrt {1+c^2 x^2}+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.92

method	result	size
default	\(a \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )+\sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \sqrt {\pi }\, b +\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -b c x \sqrt {\pi }+b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }-b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\)	\(171\)
parts	\(a \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )+\sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \sqrt {\pi }\, b +\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\, b -b c x \sqrt {\pi }+b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }-b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }\)	\(171\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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