### 3.652 $$\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{3/2} \, dx$$

Optimal. Leaf size=108 $-\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}$

[Out]

(-5*c*x*Sqrt[c - a^2*c*x^2])/8 - (5*(c - a^2*c*x^2)^(3/2))/(12*a) - ((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) -
(5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)

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Rubi [A]  time = 0.127054, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.292, Rules used = {6167, 6142, 671, 641, 195, 217, 203} $-\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*c*x*Sqrt[c - a^2*c*x^2])/8 - (5*(c - a^2*c*x^2)^(3/2))/(12*a) - ((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) -
(5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6142

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

Rule 671

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

Rule 641

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\left (c \int (1-a x)^2 \sqrt{c-a^2 c x^2} \, dx\right )\\ &=-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{4} (5 c) \int (1-a x) \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{4} (5 c) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{8} \left (5 c^2\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{1}{8} \left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.08938, size = 117, normalized size = 1.08 $\frac{c \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (6 a^4 x^4-22 a^3 x^3+25 a^2 x^2+7 a x-16\right )+30 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{24 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-16 + 7*a*x + 25*a^2*x^2 - 22*a^3*x^3 + 6*a^4*x^4) + 30*Sqrt[1 - a*x]*A
rcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(24*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.049, size = 176, normalized size = 1.6 \begin{align*}{\frac{x}{4} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,cx}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{3\,{c}^{2}}{8}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{2}{3\,a} \left ( -{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac \right ) ^{{\frac{3}{2}}}}-c\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}x-{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)+3/8*c*x*(-a^2*c*x^2+c)^(1/2)+3/8*c^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x
^2+c)^(1/2))-2/3/a*(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(3/2)-c*(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2)*x-c^2/(a^2*
c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72989, size = 412, normalized size = 3.81 \begin{align*} \left [\frac{15 \, \sqrt{-c} c \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, a}, \frac{15 \, c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, a}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*sqrt(-c)*c*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(6*a^3*c*x^3 - 16*a^2*c*x^
2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a, 1/24*(15*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x
^2 - c)) - (6*a^3*c*x^3 - 16*a^2*c*x^2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a]

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Sympy [C]  time = 9.25963, size = 340, normalized size = 3.15 \begin{align*} - a^{2} c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) - c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*c*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*
sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x*
*5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)
) + sqrt(c)*asin(a*x)/(8*a**3), True)) + 2*a*c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**
2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) - c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)
*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/
2 + sqrt(c)*asin(a*x)/(2*a), True))

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Giac [A]  time = 1.14766, size = 115, normalized size = 1.06 \begin{align*} -\frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \, a^{2} c x - 8 \, a c\right )} x + 9 \, c\right )} x + \frac{16 \, c}{a}\right )} + \frac{5 \, c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt{-c}{\left | a \right |}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(-a^2*c*x^2 + c)*((2*(3*a^2*c*x - 8*a*c)*x + 9*c)*x + 16*c/a) + 5/8*c^2*log(abs(-sqrt(-a^2*c)*x + sq
rt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a))