3.304 \(\int \frac{x (a+b \sinh ^{-1}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ -\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt{c^2 d x^2+d}} \]

[Out]

-((a + b*ArcSinh[c*x])^2/(c^2*d*Sqrt[d + c^2*d*x^2])) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^A
rcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^2
*d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2
])

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Rubi [A]  time = 0.186987, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5717, 5693, 4180, 2279, 2391} \[ -\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcSinh[c*x])^2/(c^2*d*Sqrt[d + c^2*d*x^2])) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^A
rcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^2
*d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2
])

Rule 5717

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

Rule 5693

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{c d \sqrt{d+c^2 d x^2}}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}-\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^2 d \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.454932, size = 217, normalized size = 1.15 \[ -\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+a^2-4 a b \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+2 a b \sinh ^{-1}(c x)+2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+b^2 \sinh ^{-1}(c x)^2}{c^2 d \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a^2 + 2*a*b*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 - 4*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I
)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[
1 + I/E^ArcSinh[c*x]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x
^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]))

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Maple [B]  time = 0.151, size = 446, normalized size = 2.4 \begin{align*} -{\frac{{a}^{2}}{{c}^{2}d}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{d}^{2}{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{2\,i{b}^{2}{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+i \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{2\,i{b}^{2}{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1-i \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{2\,i{b}^{2}}{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it dilog} \left ( 1+i \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{2\,i{b}^{2}}{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it dilog} \left ( 1-i \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-2\,{\frac{ab\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{2\,iab}{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1}+i \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{2\,iab}{{d}^{2}{c}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1}-i \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2}}{\sqrt{c^{2} d x^{2} + d} c^{2} d} + \int \frac{b^{2} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}} + \frac{2 \, a b x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} x \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x \operatorname{arsinh}\left (c x\right ) + a^{2} x\right )}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*x*arcsinh(c*x)^2 + 2*a*b*x*arcsinh(c*x) + a^2*x)/(c^4*d^2*x^4 + 2*c^2*d^2*x^
2 + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(3/2), 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} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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