3.1307 \(\int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{x (c-a^2 c x^2)^{9/8}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2\ 2^{5/8} \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2} F_1\left (\frac{1}{8};\frac{11}{8},1;\frac{9}{8};\frac{1}{2} (a x+1),a x+1\right )}{c \sqrt [8]{c-a^2 c x^2}} \]

[Out]

(-2*2^(5/8)*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8)*AppellF1[1/8, 11/8, 1, 9/8, (1 + a*x)/2, 1 + a*x])/(c*(c - a^2
*c*x^2)^(1/8))

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Rubi [A]  time = 0.22463, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6153, 6150, 136} \[ -\frac{2\ 2^{5/8} \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2} F_1\left (\frac{1}{8};\frac{11}{8},1;\frac{9}{8};\frac{1}{2} (a x+1),a x+1\right )}{c \sqrt [8]{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(ArcTanh[a*x]/2)/(x*(c - a^2*c*x^2)^(9/8)),x]

[Out]

(-2*2^(5/8)*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8)*AppellF1[1/8, 11/8, 1, 9/8, (1 + a*x)/2, 1 + a*x])/(c*(c - a^2
*c*x^2)^(1/8))

Rule 6153

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

Rule 6150

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

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{9/8}} \, dx &=\frac{\sqrt [8]{1-a^2 x^2} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{9/8}} \, dx}{c \sqrt [8]{c-a^2 c x^2}}\\ &=\frac{\sqrt [8]{1-a^2 x^2} \int \frac{1}{x (1-a x)^{11/8} (1+a x)^{7/8}} \, dx}{c \sqrt [8]{c-a^2 c x^2}}\\ &=-\frac{2\ 2^{5/8} \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2} F_1\left (\frac{1}{8};\frac{11}{8},1;\frac{9}{8};\frac{1}{2} (1+a x),1+a x\right )}{c \sqrt [8]{c-a^2 c x^2}}\\ \end{align*}

Mathematica [F]  time = 0.564326, size = 0, normalized size = 0. \[ \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{9/8}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(ArcTanh[a*x]/2)/(x*(c - a^2*c*x^2)^(9/8)),x]

[Out]

Integrate[E^(ArcTanh[a*x]/2)/(x*(c - a^2*c*x^2)^(9/8)), x]

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{8}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x)

[Out]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(9/8)*x), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x/(-a**2*c*x**2+c)**(9/8),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x, algorithm="giac")

[Out]

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(9/8)*x), x)