\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {d+c^2 d x^2}} \, dx\) [153]

Optimal result

Integrand size = 26, antiderivative size = 203 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {d+c^2 d x^2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{2 x \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 d x^2}+\frac {c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {d+c^2 d x^2}}-\frac {b c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {d+c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 203, 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 = {5809, 5816, 4267, 2317, 2438, 30} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\frac {c^2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 d x^2}+\frac {b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 d x^2+d}}-\frac {b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1}}{2 x \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && 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 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]

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

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\frac {-\frac {4 a \sqrt {d+c^2 d x^2}}{x^2}-4 a c^2 \sqrt {d} \log (x)+4 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 d^2 \left (1+c^2 x^2\right )^{3/2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}}{8 d} \]

[In]

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

[Out]

((-4*a*Sqrt[d + c^2*d*x^2])/x^2 - 4*a*c^2*Sqrt[d]*Log[x] + 4*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]
] + (b*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[
c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcSinh[c*x])]
+ 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(d + c^2*d
*x^2)^(3/2))/(8*d)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.64

method	result	size
default	\(-\frac {a \sqrt {c^{2} d \,x^{2}+d}}{2 d \,x^{2}}+\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(333\)
parts	\(-\frac {a \sqrt {c^{2} d \,x^{2}+d}}{2 d \,x^{2}}+\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(333\)

[In]

int((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{3} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate((a+b*asinh(c*x))/x**3/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asinh(c*x))/(x**3*sqrt(d*(c**2*x**2 + 1))), x)

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^(1/2)), x)