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

Optimal. Leaf size=167 \[ -\frac{b^2 c \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}}+\frac{2 b c \sqrt{c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.226659, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5723, 5659, 3716, 2190, 2279, 2391} \[ \frac{b^2 c \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}}+\frac{2 b c \sqrt{c^2 x^2+1} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5723

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[(b*c*n*d^IntPart[p]*(d + e
*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Arc
Sinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[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]

Rubi steps

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

Mathematica [A]  time = 0.449452, size = 168, normalized size = 1.01 \[ \frac{-b^2 c x \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )-a \left (a c^2 x^2+a-2 b c x \sqrt{c^2 x^2+1} \log (c x)\right )-2 b \sinh ^{-1}(c x) \left (a c^2 x^2+a-b c x \sqrt{c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+b^2 \left (-c^2 x^2+c x \sqrt{c^2 x^2+1}-1\right ) \sinh ^{-1}(c x)^2}{x \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.219, size = 526, normalized size = 3.2 \begin{align*} -{\frac{{a}^{2}}{dx}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x{c}^{2}}{d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}c}{d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{xd \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}}+2\,{\frac{{b}^{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}}+2\,{\frac{{b}^{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}}+2\,{\frac{{b}^{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}}-2\,{\frac{ab\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}}-2\,{\frac{ab\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) x{c}^{2}}{d \left ({c}^{2}{x}^{2}+1 \right ) }}-2\,{\frac{ab\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) }{xd \left ({c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{ab\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) c}{\sqrt{{c}^{2}{x}^{2}+1}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/d/x*(c^2*d*x^2+d)^(1/2)-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2/(c^2*x^2+1)/d*x*c^2-b^2*(d*(c^2*x^2+1))^
(1/2)*arcsinh(c*x)^2/(c^2*x^2+1)^(1/2)/d*c-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2/(c^2*x^2+1)/d/x+2*b^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*b^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*b^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*b^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*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*c-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*
x)/(c^2*x^2+1)/d*x*c^2-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/d/x+2*a*b*(d*(c^2*x^2+1))^(1/2)/(c
^2*x^2+1)^(1/2)/d*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/x^2/(c^2*d*x^2+d)^(1/2),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 )}}{c^{2} d x^{4} + d x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(1/2),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)/(c^2*d*x^4 + d*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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