3.263 \(\int \frac{\sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2}{x^2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{b^2 c \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{c^2 x^2+1}}+\frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{2 b c \sqrt{c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}} \]

[Out]

-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x) + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^
2*x^2] + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*Sqrt[1 + c^2*x^2]) + (2*b*c*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^
(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2]

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Rubi [A]  time = 0.254924, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5737, 5659, 3716, 2190, 2279, 2391, 5675} \[ \frac{b^2 c \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{c^2 x^2+1}}-\frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{2 b c \sqrt{c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x) - (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^
2*x^2] + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*Sqrt[1 + c^2*x^2]) + (2*b*c*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] + (b^2*c*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(
2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2]

Rule 5737

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*Sqrt[d + e*x^2])/(f*(m +
1)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d + e*x^2])/(f
^2*(m + 1)*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 5659

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

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 2391

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 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{1+c^2 x^2}}+\frac{\left (2 b c \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{1+c^2 x^2}}-\frac{\left (4 b c \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{1+c^2 x^2}}+\frac{2 b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{1+c^2 x^2}}+\frac{2 b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 c \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}+\frac{c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt{1+c^2 x^2}}+\frac{2 b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b^2 c \sqrt{d+c^2 d x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.15233, size = 232, normalized size = 1.11 \[ \frac{b^2 c \sqrt{c^2 d x^2+d} \left (\sinh ^{-1}(c x) \left (\left (3-\frac{3 \sqrt{c^2 x^2+1}}{c x}\right ) \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2+6 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-3 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )}{3 \sqrt{c^2 x^2+1}}-\frac{a^2 \sqrt{c^2 d x^2+d}}{x}+a^2 c \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+\frac{a b \sqrt{c^2 d x^2+d} \left (-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right )}{x \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a^2*Sqrt[d + c^2*d*x^2])/x) + (a*b*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x
]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) + a^2*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b^2*c
*Sqrt[d + c^2*d*x^2]*(ArcSinh[c*x]*((3 - (3*Sqrt[1 + c^2*x^2])/(c*x))*ArcSinh[c*x] + ArcSinh[c*x]^2 + 6*Log[1
- E^(-2*ArcSinh[c*x])]) - 3*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(3*Sqrt[1 + c^2*x^2])

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Maple [B]  time = 0.241, size = 625, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/d/x*(c^2*d*x^2+d)^(3/2)+a^2*c^2*x*(c^2*d*x^2+d)^(1/2)+a^2*c^2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1
/2))/(c^2*d)^(1/2)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^3*c-b^2*(d*(c^2*x^2+1))^(1/2)*
arcsinh(c*x)^2*x/(c^2*x^2+1)*c^2-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2/(c^2*x^2+1)^(1/2)*c-b^2*(d*(c^2*x^2+
1))^(1/2)*arcsinh(c*x)^2/x/(c^2*x^2+1)+2*b^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*b^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*b^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*b^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+a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c
-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)*x/(c^2*
x^2+1)*c^2-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/x/(c^2*x^2+1)+2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2
)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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