\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)} \, dx\) [33]

Optimal result

Integrand size = 24, antiderivative size = 61 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5799, 5569, 4267, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]

[In]

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

[Out]

(-2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d - (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d) + (b*PolyL
og[2, E^(2*ArcSinh[c*x])])/(2*d)

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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5799

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(61)=122\).

Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.39 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {a \text {arcsinh}(c x)}{d}-\frac {b \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {a \log \left (1+c^2 x^2\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]

[In]

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

[Out]

-((a*ArcSinh[c*x])/d) - (b*ArcSinh[c*x]*Log[1 - (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d - (b*ArcSinh[c*x]*Log[1 + (S
qrt[-c^2]*E^ArcSinh[c*x])/c])/d + (a*Log[1 - E^(2*ArcSinh[c*x])])/d + (b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x
])])/d - (a*Log[1 + c^2*x^2])/(2*d) - (b*PolyLog[2, -((Sqrt[-c^2]*E^ArcSinh[c*x])/c)])/d - (b*PolyLog[2, (Sqrt
[-c^2]*E^ArcSinh[c*x])/c])/d + (b*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.64

method	result	size
parts	\(\frac {a \left (-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (x \right )\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(161\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(163\)
default	\(\frac {a \left (\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(163\)

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a}{c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{3} + x}\, dx}{d} \]

[In]

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

[Out]

(Integral(a/(c**2*x**3 + x), x) + Integral(b*asinh(c*x)/(c**2*x**3 + x), x))/d

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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