3.248 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}} \]

[Out]

(3*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x]) + (c^2*(1 - a^2*x^2)^(3/2))/(a*(c - a*c*x)^(5/2)) - (3*Sqrt[2]*ArcTa
nh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*Sqrt[c])

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Rubi [A]  time = 0.100717, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 663, 665, 661, 208} \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x]) + (c^2*(1 - a^2*x^2)^(3/2))/(a*(c - a*c*x)^(5/2)) - (3*Sqrt[2]*ArcTa
nh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*Sqrt[c])

Rule 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rule 663

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1
, 0] && IntegerQ[2*p]

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 661

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{7/2}} \, dx\\ &=\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac{1}{2} (3 c) \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac{3 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-3 \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+(6 a c) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ &=\frac{3 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}}\\ \end{align*}

Mathematica [C]  time = 0.022388, size = 57, normalized size = 0.5 \[ \frac{(a x+1)^{5/2} (c-a c x)^{3/2} \text{Hypergeometric2F1}\left (2,\frac{5}{2},\frac{7}{2},\frac{1}{2} (a x+1)\right )}{10 a c^2 (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + a*x)^(5/2)*(c - a*c*x)^(3/2)*Hypergeometric2F1[2, 5/2, 7/2, (1 + a*x)/2])/(10*a*c^2*(1 - a*x)^(3/2))

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Maple [A]  time = 0.105, size = 127, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( ax-1 \right ) ^{2}a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-2\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c+4\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x)

[Out]

(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(3*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c-2*x*a*(c
*(a*x+1))^(1/2)*c^(1/2)-3*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c+4*(c*(a*x+1))^(1/2)*c^(1/2)
)/c^(3/2)/(a*x-1)^2/(c*(a*x+1))^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.2702, size = 608, normalized size = 5.29 \begin{align*} \left [-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x - 2\right )} - \frac{3 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \log \left (-\frac{a^{2} x^{2} + 2 \, a x + \frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt{c}}}{2 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac{3 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a^{2} x^{2} - 1}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x - 2\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*(4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 2) - 3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*log(-(a^2*x^2 + 2
*a*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a^2*x^2 - 2*a*x + 1))/sqrt(c))/(a^3*c*x^2 -
2*a^2*c*x + a*c), -(3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*
c*x + c)*sqrt(-1/c)/(a^2*x^2 - 1)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 2))/(a^3*c*x^2 - 2*a^2*c*x +
a*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26486, size = 95, normalized size = 0.83 \begin{align*} \frac{\frac{3 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + 2 \, \sqrt{a c x + c} - \frac{2 \, \sqrt{a c x + c} c}{a c x - c}}{a{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x, algorithm="giac")

[Out]

(3*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + 2*sqrt(a*c*x + c) - 2*sqrt(a*c*x + c)*c/(
a*c*x - c))/(a*abs(c))