3.1293 \(\int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac {2 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{3 a c \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6135} \[ -\frac {2 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{3 a c \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6135

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n - a*x)*E^(n*ArcTanh[a*x]))
/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 64, normalized size = 1.56 \[ \frac {2 (2 a x-1) \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/4} \sqrt [4]{a x+1} \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 55, normalized size = 1.34 \[ -\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2 a x -1\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{3 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 1.26, size = 49, normalized size = 1.20 \[ \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {4\,x}{3\,c}-\frac {2}{3\,a\,c}\right )}{\sqrt {c-a^2\,c\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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