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

Optimal. Leaf size=90 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{2 \sqrt{2} a c^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{2 a c (c-a c x)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0822888, antiderivative size = 90, 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, 673, 661, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{2 \sqrt{2} a c^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{2 a c (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - a^2*x^2]/(2*a*c*(c - a*c*x)^(3/2)) + ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(
2*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 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[m, 0] && NeQ[m + 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^{-\tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac{\int \frac{1}{(c-a c x)^{3/2} \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 a c (c-a c x)^{3/2}}+\frac{\int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{4 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 a c (c-a c x)^{3/2}}-\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 )}{2 c}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 a c (c-a c x)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{2 \sqrt{2} a c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0584424, size = 70, normalized size = 0.78 \[ -\frac{\sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )-2 \sqrt{a x+1}}{4 a c^2 \sqrt{1-a x} \sqrt{c-a c x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.103, size = 111, normalized size = 1.2 \begin{align*} -{\frac{1}{4\, \left ( ax-1 \right ) ^{2}a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac-\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) c-2\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)/c^(7/2)*(2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a
*c-2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c-2*(c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)^2/(c*(a*x+1))
^(1/2)/a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.24517, size = 589, normalized size = 6.54 \begin{align*} \left [\frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 ) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{8 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(sqrt(2)*(a^2*x^2 - 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)*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 - 2*a^2*c^
3*x + a*c^3), 1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*s
qrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1))**(5/2)*(a*x + 1)), x)

________________________________________________________________________________________

Giac [A]  time = 1.51033, size = 92, normalized size = 1.02 \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{2 \, \sqrt{a c x + c}}{{\left (a c x - c\right )} a c}\right )}{\left | c \right |}}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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