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

Optimal. Leaf size=119 \[ -\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a} \]

[Out]

(-4*c*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x]) - (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2)) + (4*Sqrt[2
]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/a

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Rubi [A]  time = 0.103845, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6127, 665, 661, 208} \[ -\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x]) - (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2)) + (4*Sqrt[2
]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/a

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 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 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)^{5/2}} \, dx\\ &=-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\left (2 c^2\right ) \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+(4 c) \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\left (8 a c^2\right ) \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{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0326463, size = 67, normalized size = 0.56 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1} (a x+7)-6 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{3 a \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(7 + a*x) - 6*Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/(3*a*Sqrt[1 - a*x])

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Maple [A]  time = 0.096, size = 95, normalized size = 0.8 \begin{align*} -{\frac{2}{ \left ( 3\,ax-3 \right ) a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 6\,\sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -xa\sqrt{c \left ( ax+1 \right ) }-7\,\sqrt{c \left ( ax+1 \right ) } \right ){\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]

-2/3*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(6*c^(1/2)*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-x
*a*(c*(a*x+1))^(1/2)-7*(c*(a*x+1))^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2)/a

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

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Fricas [A]  time = 2.2336, size = 518, normalized size = 4.35 \begin{align*} \left [\frac{2 \,{\left (3 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 7\right )}\right )}}{3 \,{\left (a^{2} x - a\right )}}, \frac{2 \,{\left (6 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 7\right )}\right )}}{3 \,{\left (a^{2} x - a\right )}}\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]

[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sq
rt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a), 2/3*(6*sqrt(
2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2
*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a)]

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

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Giac [A]  time = 1.26187, size = 142, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (\frac{6 \, \sqrt{2} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} +{\left (a c x + c\right )}^{\frac{3}{2}} + 6 \, \sqrt{a c x + c} c\right )}}{3 \, a{\left | c \right |}} + \frac{4 \, \sqrt{2}{\left (3 \, c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + 4 \, \sqrt{-c} c^{\frac{3}{2}}\right )}}{3 \, a \sqrt{-c}{\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]

-2/3*(6*sqrt(2)*c^2*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + (a*c*x + c)^(3/2) + 6*sqrt(a*c*x +
c)*c)/(a*abs(c)) + 4/3*sqrt(2)*(3*c^2*arctan(sqrt(c)/sqrt(-c)) + 4*sqrt(-c)*c^(3/2))/(a*sqrt(-c)*abs(c))