\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)^2} \, dx\) [42]

Optimal result

Integrand size = 24, antiderivative size = 110 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5811, 5799, 5569, 4267, 2317, 2438, 197} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (c^2 x^2+1\right )}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b c x}{2 d^2 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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]

Rule 5811

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d +
e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d} \\ & = -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {2 \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \\ & = -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(110)=220\).

Time = 0.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {\frac {a^2}{b}-\frac {a}{1+c^2 x^2}+\frac {b c x}{\sqrt {1+c^2 x^2}}+2 a \text {arcsinh}(c x)-\frac {b \text {arcsinh}(c x)}{1+c^2 x^2}+2 b \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-2 a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-2 b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+a \log \left (1+c^2 x^2\right )+2 b \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97

method	result	size
derivativedivides	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}+2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(217\)
default	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}+2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(217\)
parts	\(\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {a \ln \left (x \right )}{d^{2}}+\frac {b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(222\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**5 + 2*c**2*x**3 + x), x) + Integral(b*asinh(c*x)/(c**4*x**5 + 2*c**2*x**3 + x), x))/d**2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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