3.275 \(\int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{\sqrt{2} a c^{5/2}}-\frac{\sqrt{c-a c x}}{a c^3 \sqrt{1-a^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0810259, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6127, 667, 661, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{\sqrt{2} a c^{5/2}}-\frac{\sqrt{c-a c x}}{a c^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(5/2)),x]

[Out]

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

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 667

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(d*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*a*e*(p + 1)), x] + Dist[(d*(m + 2*p + 2))/(2*a*(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] && LtQ[p, -1] && LtQ[0, m, 1] && 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)}}{(c-a c x)^{5/2}} \, dx &=\frac{\int \frac{\sqrt{c-a c x}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{\sqrt{c-a c x}}{a c^3 \sqrt{1-a^2 x^2}}+\frac{\int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{2 c^2}\\ &=-\frac{\sqrt{c-a c x}}{a c^3 \sqrt{1-a^2 x^2}}-\frac{a \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 )}{c}\\ &=-\frac{\sqrt{c-a c x}}{a c^3 \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{\sqrt{2} a c^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0243918, size = 55, normalized size = 0.65 \[ -\frac{(1-a x)^{3/2} \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{2} (a x+1)\right )}{a c \sqrt{a x+1} (c-a c x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(5/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.102, size = 82, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ({\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) \sqrt{2}\sqrt{c \left ( ax+1 \right ) }-2\,\sqrt{c} \right ){c}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (-a c x + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((-a*c*x + c)^(5/2)*(a*x + 1)^3), x)

________________________________________________________________________________________

Fricas [A]  time = 1.63432, size = 529, normalized size = 6.22 \begin{align*} \left [\frac{\sqrt{2}{\left (a^{2} x^{2} - 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 ) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \,{\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}, \frac{\sqrt{2}{\left (a^{2} x^{2} - 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 ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \,{\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(2)*(a^2*x^2 - 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)*
sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - a*c^3), 1/2*(sqr
t(2)*(a^2*x^2 - 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)) + 2*s
qrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - a*c^3)]

________________________________________________________________________________________

Sympy [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(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(5/2),x)

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [A]  time = 1.19903, size = 127, normalized size = 1.49 \begin{align*} -\frac{{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c} c} - \frac{\sqrt{2}{\left (\sqrt{c} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{-c}\right )}}{a \sqrt{-c} c^{\frac{3}{2}}} + \frac{2}{\sqrt{a c x + c} a c}\right )}{\left | c \right |}}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="giac")

[Out]

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