### 3.636 $$\int e^{3 \coth ^{-1}(a x)} (c-a^2 c x^2)^{3/2} \, dx$$

Optimal. Leaf size=46 $\frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.176097, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {6192, 6193, 32} $\frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6192

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

Rule 6193

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{3/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (c-a^2 c x^2\right )^{3/2} \int (1+a x)^3 \, dx}{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{(1+a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}

Mathematica [A]  time = 0.0313415, size = 58, normalized size = 1.26 $-\frac{c \left (a^3 x^3+4 a^2 x^2+6 a x+4\right ) \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*Sqrt[c - a^2*c*x^2]*(4 + 6*a*x + 4*a^2*x^2 + a^3*x^3))/(4*a*Sqrt[1 - 1/(a^2*x^2)])

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Maple [A]  time = 0.116, size = 60, normalized size = 1.3 \begin{align*}{\frac{x \left ({x}^{3}{a}^{3}+4\,{a}^{2}{x}^{2}+6\,ax+4 \right ) }{4\, \left ( ax+1 \right ) ^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.10909, size = 131, normalized size = 2.85 \begin{align*} -\frac{{\left (a^{5} \sqrt{-c} c x^{5} + 3 \, a^{4} \sqrt{-c} c x^{4} + 2 \, a^{3} \sqrt{-c} c x^{3} - 2 \, a^{2} \sqrt{-c} c x^{2} - 4 \, \sqrt{-c} c\right )}{\left (a x + 1\right )}^{2}}{4 \,{\left (a^{3} x^{2} + 2 \, a^{2} x + a\right )}{\left (a x - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a^5*sqrt(-c)*c*x^5 + 3*a^4*sqrt(-c)*c*x^4 + 2*a^3*sqrt(-c)*c*x^3 - 2*a^2*sqrt(-c)*c*x^2 - 4*sqrt(-c)*c)*
(a*x + 1)^2/((a^3*x^2 + 2*a^2*x + a)*(a*x - 1))

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Fricas [A]  time = 1.59019, size = 90, normalized size = 1.96 \begin{align*} -\frac{{\left (a^{3} c x^{4} + 4 \, a^{2} c x^{3} + 6 \, a c x^{2} + 4 \, c x\right )} \sqrt{-a^{2} c}}{4 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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