\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (d+c^2 d x^2)} \, dx\) [35]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 113, 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 = {5809, 5799, 5569, 4267, 2317, 2438, 270} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {2 c^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b c \sqrt {c^2 x^2+1}}{2 d x} \]

[In]

Int[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)),x]

[Out]

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

Rule 270

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

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 5809

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/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{2 d x^2}-c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {1}{x^2 \sqrt {1+c^2 x^2}} \, dx}{2 d} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}-\frac {c^2 \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}-\frac {\left (2 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b c^2 \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. \(240\) vs. \(2(113)=226\).

Time = 0.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {-\frac {b c \sqrt {1+c^2 x^2}}{x}-b c^2 \text {arcsinh}(c x)^2-\frac {a+b \text {arcsinh}(c x)}{x^2}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{b}+2 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+a c^2 \log \left (1+c^2 x^2\right )+2 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{2 d} \]

[In]

Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.92

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

[In]

int((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x,method=_RETURNVERBOSE)

[Out]

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

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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