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]

-2/195*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-14*a*x+1)/a/c/(-a^2*c*x^2+c)^(7/2)-112/6435*((a*x+1)/(-a^2*x^2+1)^
(1/2))^(1/2)*(-10*a*x+1)/a/c^2/(-a^2*c*x^2+c)^(5/2)-256/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-6*a*x+1)/a/c
^3/(-a^2*c*x^2+c)^(3/2)-2048/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-2*a*x+1)/a/c^4/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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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)^(9/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]

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)^(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)

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maple [A]  time = 0.03, size = 103, normalized size = 0.62 \[ \frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 x^{5} a^{5}+3200 x^{4} a^{4}+8240 x^{3} a^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}} \]

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)^(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 = 0.00, 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)^(9/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.49, size = 183, normalized size = 1.11 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {2482}{6435\,a^7\,c^4}+\frac {4096\,x^7}{6435\,c^4}-\frac {7676\,x}{6435\,a^6\,c^4}-\frac {2048\,x^6}{6435\,a\,c^4}-\frac {1536\,x^5}{715\,a^2\,c^4}+\frac {1280\,x^4}{1287\,a^3\,c^4}+\frac {3296\,x^3}{1287\,a^4\,c^4}-\frac {752\,x^2}{715\,a^5\,c^4}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}} \]

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)^(9/2),x)

[Out]

-(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)*(2482/(6435*a^7*c^4) + (4096*x^7)/(6435*c^4) - (7676*x)/(6435*a^6*c^4)
- (2048*x^6)/(6435*a*c^4) - (1536*x^5)/(715*a^2*c^4) + (1280*x^4)/(1287*a^3*c^4) + (3296*x^3)/(1287*a^4*c^4)
- (752*x^2)/(715*a^5*c^4)))/((c - a^2*c*x^2)^(1/2)/a^6 - x^6*(c - a^2*c*x^2)^(1/2) + (3*x^4*(c - a^2*c*x^2)^(1
/2))/a^2 - (3*x^2*(c - a^2*c*x^2)^(1/2))/a^4)

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sympy [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)**(9/2),x)

[Out]

Timed out

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