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

Optimal. Leaf size=82 \[ \frac {\sqrt {1-a^2 x^2}}{a (c-a c x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{\sqrt {2} a c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6127, 663, 661, 208} \[ \frac {\sqrt {1-a^2 x^2}}{a (c-a c x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{\sqrt {2} a c^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

Rule 661

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]

Rule 663

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

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

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Mathematica [A]  time = 0.06, size = 80, normalized size = 0.98 \[ -\frac {\sqrt {c-a c x} \left (2 a x+(a x-1) \sqrt {2 a x+2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )+2\right )}{2 a c^2 (a x-1) \sqrt {1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.67, size = 258, normalized size = 3.15 \[ \left [\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c
*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^2*x^2 - 2*a^2*c^
2*x + a*c^2), -1/2*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*
sqrt(-c)/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

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giac [A]  time = 0.33, size = 58, normalized size = 0.71 \[ \frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, \sqrt {a c x + c}}{a c x - c}}{2 \, a {\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) - 2*sqrt(a*c*x + c)/(a*c*x - c))/(a*abs(c))

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maple [A]  time = 0.05, size = 111, normalized size = 1.35 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c -\sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{2 \left (a x -1\right )^{2} \sqrt {c \left (a x +1\right )}\, c^{\frac {5}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c-2^(1/2
)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c+2*(c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)^2/(c*(a*x+1))^(1/2)/c^
(5/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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