3.5.66 \(\int \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x) \, dx\) [466]

Optimal. Leaf size=235 \[ \frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)-\frac {c \sqrt {1-a^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a \sqrt {c-a^2 c x^2}}-\frac {i c \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}} \]

[Out]

-c*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(1/2)-1/2*I*c*polylog
(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(1/2)+1/2*I*c*polylog(2,I*(-a*x+1)^(1/
2)/(a*x+1)^(1/2))*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(1/2)+1/2*(-a^2*c*x^2+c)^(1/2)/a+1/2*x*arctanh(a*x)*(-a^
2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6089, 6101, 6097} \begin {gather*} -\frac {c \sqrt {1-a^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a \sqrt {c-a^2 c x^2}}-\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcTanh[a*x],x]

[Out]

Sqrt[c - a^2*c*x^2]/(2*a) + (x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x])/2 - (c*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]
/Sqrt[1 + a*x]]*ArcTanh[a*x])/(a*Sqrt[c - a^2*c*x^2]) - ((I/2)*c*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]) + ((I/2)*c*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 6089

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 6097

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]

Rule 6101

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]

Rubi steps

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

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Mathematica [A]
time = 0.22, size = 119, normalized size = 0.51 \begin {gather*} \frac {\sqrt {c \left (1-a^2 x^2\right )} \left (1+a x \tanh ^{-1}(a x)-\frac {i \left (\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcTanh[a*x],x]

[Out]

(Sqrt[c*(1 - a^2*x^2)]*(1 + a*x*ArcTanh[a*x] - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTa
nh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)

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Maple [A]
time = 1.83, size = 319, normalized size = 1.36

method	result	size
default	\(\frac {\left (a x \arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \arctanh \left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \arctanh \left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)*arctanh(a*x),x,method=_RETURNVERBOSE)

[Out]

1/2*(a*x*arctanh(a*x)+1)*(-(a*x-1)*(a*x+1)*c)^(1/2)/a+1/2*I/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))-1/2*I/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))+1/2*I/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))-1/2*I/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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arctanh(a*x),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arctanh(a*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*atanh(a*x), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arctanh(a*x),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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \mathrm {atanh}\left (a\,x\right )\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(atanh(a*x)*(c - a^2*c*x^2)^(1/2), x)

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