3.1296 \(\int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{9/2}} \, dx\)

Optimal. Leaf size=165 \[ -\frac{2048 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^4 \sqrt{c-a^2 c x^2}}-\frac{256 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{112 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-14 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{195 a c \left (c-a^2 c x^2\right )^{7/2}} \]

[Out]

(-2*E^(ArcTanh[a*x]/2)*(1 - 14*a*x))/(195*a*c*(c - a^2*c*x^2)^(7/2)) - (112*E^(ArcTanh[a*x]/2)*(1 - 10*a*x))/(
6435*a*c^2*(c - a^2*c*x^2)^(5/2)) - (256*E^(ArcTanh[a*x]/2)*(1 - 6*a*x))/(6435*a*c^3*(c - a^2*c*x^2)^(3/2)) -
(2048*E^(ArcTanh[a*x]/2)*(1 - 2*a*x))/(6435*a*c^4*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.197033, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6136, 6135} \[ -\frac{2048 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^4 \sqrt{c-a^2 c x^2}}-\frac{256 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{112 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-14 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{195 a c \left (c-a^2 c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*E^(ArcTanh[a*x]/2)*(1 - 14*a*x))/(195*a*c*(c - a^2*c*x^2)^(7/2)) - (112*E^(ArcTanh[a*x]/2)*(1 - 10*a*x))/(
6435*a*c^2*(c - a^2*c*x^2)^(5/2)) - (256*E^(ArcTanh[a*x]/2)*(1 - 6*a*x))/(6435*a*c^3*(c - a^2*c*x^2)^(3/2)) -
(2048*E^(ArcTanh[a*x]/2)*(1 - 2*a*x))/(6435*a*c^4*Sqrt[c - a^2*c*x^2])

Rule 6136

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^2
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p
, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]

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 )^{9/2}} \, dx &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}+\frac{56 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{65 c}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{896 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{1287 c^2}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{256 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{1024 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{2145 c^3}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{256 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2048 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{6435 a c^4 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0527619, size = 112, normalized size = 0.68 \[ -\frac{2 \sqrt{1-a^2 x^2} \left (2048 a^7 x^7-1024 a^6 x^6-6912 a^5 x^5+3200 a^4 x^4+8240 a^3 x^3-3384 a^2 x^2-3838 a x+1241\right )}{6435 a c^4 (1-a x)^{15/4} (a x+1)^{13/4} \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2]*(1241 - 3838*a*x - 3384*a^2*x^2 + 8240*a^3*x^3 + 3200*a^4*x^4 - 6912*a^5*x^5 - 1024*a^6*
x^6 + 2048*a^7*x^7))/(6435*a*c^4*(1 - a*x)^(15/4)*(1 + a*x)^(13/4)*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.031, size = 103, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) \left ( 2048\,{a}^{7}{x}^{7}-1024\,{x}^{6}{a}^{6}-6912\,{x}^{5}{a}^{5}+3200\,{x}^{4}{a}^{4}+8240\,{x}^{3}{a}^{3}-3384\,{a}^{2}{x}^{2}-3838\,ax+1241 \right ) }{6435\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{2}}}} \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)/(-a^2*c*x^2+c)^(9/2),x)

[Out]

2/6435*(a*x-1)*(a*x+1)*(2048*a^7*x^7-1024*a^6*x^6-6912*a^5*x^5+3200*a^4*x^4+8240*a^3*x^3-3384*a^2*x^2-3838*a*x
+1241)*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/a/(-a^2*c*x^2+c)^(9/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)/(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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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)/(-a^2*c*x^2+c)^(9/2),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)/(-a**2*c*x**2+c)**(9/2),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}{2}}}\,{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)/(-a^2*c*x^2+c)^(9/2),x, algorithm="giac")

[Out]

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