3.273 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\)

Optimal. Leaf size=67 \[ \frac {2 (c-a c x)^{3/2}}{a c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {c-a c x}}{a c \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {2 (c-a c x)^{3/2}}{a c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {c-a c x}}{a c \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[c - a*c*x])/(a*c*Sqrt[1 - a^2*x^2]) + (2*(c - a*c*x)^(3/2))/(a*c^2*Sqrt[1 - a^2*x^2])

Rule 649

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 657

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*Simplify[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, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 40, normalized size = 0.60 \[ -\frac {2 \sqrt {1-a x} (a x+3)}{a \sqrt {a x+1} \sqrt {c-a c x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 43, normalized size = 0.64 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 3\right )}}{a^{3} c x^{2} - a c} \]

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)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 50, normalized size = 0.75 \[ -2 \, {\left (\frac {\sqrt {a c x + c}}{a c^{2}} + \frac {2}{\sqrt {a c x + c} a c}\right )} {\left | c \right |} + \frac {4 \, \sqrt {2} {\left | c \right |}}{a c^{\frac {3}{2}}} \]

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)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 46, normalized size = 0.69 \[ \frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +3\right )}{\sqrt {-a c x +c}\, \left (a x +1\right )^{2} \left (a x -1\right ) a} \]

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)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.35, size = 30, normalized size = 0.45 \[ -\frac {2 \, {\left (a x + 3\right )} \sqrt {a x + 1}}{a^{2} \sqrt {c} x + a \sqrt {c}} \]

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)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.96, size = 64, normalized size = 0.96 \[ -\frac {\left (\frac {6\,\sqrt {1-a^2\,x^2}}{a^3\,c}+\frac {2\,x\,\sqrt {1-a^2\,x^2}}{a^2\,c}\right )\,\sqrt {c-a\,c\,x}}{\frac {1}{a^2}-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}\, dx \]

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)**(1/2),x)

[Out]

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

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