\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\) [126]

Optimal result

Integrand size = 26, antiderivative size = 201 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 201, 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 = {5805, 30, 5816, 4267, 2317, 2438} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {c^2 \sqrt {c^2 d x^2+d} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {b c^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {b c^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d}}{2 x \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

-1/2*(b*c*Sqrt[d + c^2*d*x^2])/(x*Sqrt[1 + c^2*x^2]) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*x^2) - (c
^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*c^2*Sqrt[d + c^2*d
*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (b*c^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x
]])/(2*Sqrt[1 + c^2*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 5805

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 + 1))), x] + (-Dist[b*c*(n/(f*(m + 1)))*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] - Dist[(c^2/(f^2*(m + 1))
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(f*x)^(m + 2)*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x
]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[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 x^2}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{x^2} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 a \sqrt {d+c^2 d x^2}}{x^2}+4 a c^2 \sqrt {d} \log (x)-4 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 \sqrt {d+c^2 d x^2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}\right ) \]

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^3,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*Sqrt[d + c^2*d*x^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]))/Sqrt[1 + c^2*x
^2])/8

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.66

method	result	size
default	\(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(334\)
parts	\(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(334\)

[In]

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

[Out]

a*(-1/2/d/x^2*(c^2*d*x^2+d)^(3/2)+1/2*c^2*((c^2*d*x^2+d)^(1/2)-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/(c^2*x^2+1)-1
/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*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)*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)*arcsin
h(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)*polylog(2,c*x+(c^2*x^2+1)^(
1/2))*c^2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[In]

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

[Out]

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