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

Optimal. Leaf size=107 \[ -\frac {5 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}-\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]

[Out]

5/12*(-a^2*c*x^2+c)^(3/2)/a+1/4*(a*x+1)*(-a^2*c*x^2+c)^(3/2)/a-5/8*c^(3/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(
1/2))/a-5/8*c*x*(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, 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, 6141, 671, 641, 195, 217, 203} \[ -\frac {5 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}-\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]

Antiderivative was successfully verified.

[In]

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

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 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 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 6141

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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]) && IGt
Q[n/2, 0]

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]

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*}

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Mathematica [A]  time = 0.10, size = 117, normalized size = 1.09 \[ \frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {a x+1} \left (6 a^4 x^4+10 a^3 x^3-7 a^2 x^2-25 a x+16\right )+30 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(16 - 25*a*x - 7*a^2*x^2 + 10*a^3*x^3 + 6*a^4*x^4) + 30*Sqrt[1 - a*x]*Ar
cSin[Sqrt[1 - a*x]/Sqrt[2]]))/(24*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.60, size = 180, normalized size = 1.68 \[ \left [\frac {15 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, a}, \frac {15 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(3/2),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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giac [A]  time = 0.16, size = 85, normalized size = 0.79 \[ -\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, a^{2} c x + 8 \, a c\right )} x + 9 \, c\right )} x - \frac {16 \, c}{a}\right )} + \frac {5 \, c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt {-c} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.04, size = 188, normalized size = 1.76 \[ \frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c x \sqrt {-a^{2} c \,x^{2}+c}}{8}+\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 \sqrt {a^{2} c}}+\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a}-c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x -\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.42, size = 131, normalized size = 1.22 \[ \frac {1}{4} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x - \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x + \frac {3}{8} \, \sqrt {-a^{2} c x^{2} + c} c x - \frac {c^{3} \arcsin \left (a x - 2\right )}{a \left (-c\right )^{\frac {3}{2}}} + \frac {3 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{8 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a} + \frac {2 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 7.80, size = 342, normalized size = 3.20 \[ - a^{2} c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - 2 a c \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) - c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {\sqrt {c} x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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