3.465 \(\int (c-a^2 c x^2)^{3/2} \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=291 \[ -\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{8 a \sqrt{c-a^2 c x^2}}+\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{8 a \sqrt{c-a^2 c x^2}}-\frac{3 c^2 \sqrt{1-a^2 x^2} \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt{c-a^2 c x^2}}+\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x) \]

[Out]

(3*c*Sqrt[c - a^2*c*x^2])/(8*a) + (c - a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x])/8 +
(x*(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x])/4 - (3*c^2*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTan
h[a*x])/(4*a*Sqrt[c - a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 +
a*x]])/(a*Sqrt[c - a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])
/(a*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.149635, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5942, 5954, 5950} \[ -\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{8 a \sqrt{c-a^2 c x^2}}+\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{8 a \sqrt{c-a^2 c x^2}}-\frac{3 c^2 \sqrt{1-a^2 x^2} \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt{c-a^2 c x^2}}+\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x],x]

[Out]

(3*c*Sqrt[c - a^2*c*x^2])/(8*a) + (c - a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x])/8 +
(x*(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x])/4 - (3*c^2*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTan
h[a*x])/(4*a*Sqrt[c - a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 +
a*x]])/(a*Sqrt[c - a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])
/(a*Sqrt[c - a^2*c*x^2])

Rule 5942

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

Rule 5954

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 - c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcTanh[c*x])^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d
+ e, 0] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 5950

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

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac{1}{4} (3 c) \int \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x) \, dx\\ &=\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tanh ^{-1}(a x)}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac{\left (3 c^2 \sqrt{1-a^2 x^2}\right ) \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 \sqrt{c-a^2 c x^2}}\\ &=\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)-\frac{3 c^2 \sqrt{1-a^2 x^2} \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt{c-a^2 c x^2}}-\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{8 a \sqrt{c-a^2 c x^2}}+\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{8 a \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.653791, size = 206, normalized size = 0.71 \[ -\frac{c \sqrt{c-a^2 c x^2} \left (9 i \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-9 i \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+2 a^2 x^2 \sqrt{1-a^2 x^2}-11 \sqrt{1-a^2 x^2}+6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-15 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+9 i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-9 i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )}{24 a \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x],x]

[Out]

-(c*Sqrt[c - a^2*c*x^2]*(-11*Sqrt[1 - a^2*x^2] + 2*a^2*x^2*Sqrt[1 - a^2*x^2] - 15*a*x*Sqrt[1 - a^2*x^2]*ArcTan
h[a*x] + 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (9*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - (9*I)*ArcTa
nh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + (9*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (9*I)*PolyLog[2, I/E^ArcTanh[a*x]]
))/(24*a*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.319, size = 345, normalized size = 1.2 \begin{align*} -{\frac{c \left ( 6\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) +2\,{a}^{2}{x}^{2}-15\,ax{\it Artanh} \left ( ax \right ) -11 \right ) }{24\,a}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}+{\frac{{\frac{3\,i}{8}}c{\it Artanh} \left ( ax \right ) }{a \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{3\,i}{8}}c{\it Artanh} \left ( ax \right ) }{a \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{3\,i}{8}}c}{a \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{3\,i}{8}}c}{a \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)*arctanh(a*x),x)

[Out]

-1/24*c/a*(-(a*x-1)*(a*x+1)*c)^(1/2)*(6*a^3*x^3*arctanh(a*x)+2*a^2*x^2-15*a*x*arctanh(a*x)-11)+3/8*I*c/a/(a*x+
1)*(-a^2*x^2+1)^(1/2)/(a*x-1)*(-(a*x-1)*(a*x+1)*c)^(1/2)*arctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))-3/8*I
*c/a/(a*x+1)*(-a^2*x^2+1)^(1/2)/(a*x-1)*(-(a*x-1)*(a*x+1)*c)^(1/2)*arctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1
/2))+3/8*I*c/a/(a*x+1)*(-a^2*x^2+1)^(1/2)/(a*x-1)*(-(a*x-1)*(a*x+1)*c)^(1/2)*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1
/2))-3/8*I*c/a/(a*x+1)*(-a^2*x^2+1)^(1/2)/(a*x-1)*(-(a*x-1)*(a*x+1)*c)^(1/2)*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1
/2))

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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^2*c*x^2+c)^(3/2)*arctanh(a*x),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 (-{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} c x^{2} + c} \operatorname{artanh}\left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(3/2)*arctanh(a*x),x, algorithm="fricas")

[Out]

integral(-(a^2*c*x^2 - c)*sqrt(-a^2*c*x^2 + c)*arctanh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(3/2)*atanh(a*x),x)

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(3/2)*arctanh(a*x),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)*arctanh(a*x), x)