3.1172 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=278 \[ \frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{12 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c^3 (1-a x)^4 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt{c-a^2 c x^2}} \]

[Out]

Sqrt[1 - a^2*x^2]/(16*a*c^3*(1 - a*x)^4*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(12*a*c^3*(1 - a*x)^3*Sqrt[c
- a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^3*
(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*
x^2]*ArcTanh[a*x])/(32*a*c^3*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.138623, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6143, 6140, 44, 207} \[ \frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{12 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c^3 (1-a x)^4 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

Sqrt[1 - a^2*x^2]/(16*a*c^3*(1 - a*x)^4*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(12*a*c^3*(1 - a*x)^3*Sqrt[c
- a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^3*
(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*
x^2]*ArcTanh[a*x])/(32*a*c^3*Sqrt[c - a^2*c*x^2])

Rule 6143

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

Rule 6140

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{(1-a x)^5 (1+a x)^2} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (-\frac{1}{4 (-1+a x)^5}+\frac{1}{4 (-1+a x)^4}-\frac{3}{16 (-1+a x)^3}+\frac{1}{8 (-1+a x)^2}+\frac{1}{32 (1+a x)^2}-\frac{5}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{16 a c^3 (1-a x)^4 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{12 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (5 \sqrt{1-a^2 x^2}\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{32 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{16 a c^3 (1-a x)^4 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{12 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1+a x) \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{32 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0768633, size = 101, normalized size = 0.36 \[ \frac{\sqrt{1-a^2 x^2} \left (-15 a^4 x^4+45 a^3 x^3-35 a^2 x^2-15 a x+15 (a x-1)^4 (a x+1) \tanh ^{-1}(a x)+32\right )}{96 a c^3 (a x-1)^4 (a x+1) \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

(Sqrt[1 - a^2*x^2]*(32 - 15*a*x - 35*a^2*x^2 + 45*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^4*(1 + a*x)*ArcTanh[a*x
]))/(96*a*c^3*(-1 + a*x)^4*(1 + a*x)*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.099, size = 238, normalized size = 0.9 \begin{align*} -{\frac{15\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-15\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-45\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+45\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-30\,{x}^{4}{a}^{4}+30\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -30\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+90\,{x}^{3}{a}^{3}+30\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-30\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-70\,{a}^{2}{x}^{2}-45\,ax\ln \left ( ax+1 \right ) +45\,\ln \left ( ax-1 \right ) xa-30\,ax+15\,\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) +64}{ \left ( 192\,{a}^{2}{x}^{2}-192 \right ){c}^{4}a \left ( ax+1 \right ) \left ( ax-1 \right ) ^{4}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x)

[Out]

-1/192*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(15*ln(a*x+1)*x^5*a^5-15*ln(a*x-1)*x^5*a^5-45*ln(a*x+1)*a^4*x
^4+45*ln(a*x-1)*a^4*x^4-30*x^4*a^4+30*a^3*x^3*ln(a*x+1)-30*ln(a*x-1)*x^3*a^3+90*x^3*a^3+30*ln(a*x+1)*a^2*x^2-3
0*ln(a*x-1)*a^2*x^2-70*a^2*x^2-45*a*x*ln(a*x+1)+45*ln(a*x-1)*x*a-30*a*x+15*ln(a*x+1)-15*ln(a*x-1)+64)/(a^2*x^2
-1)/c^4/a/(a*x+1)/(a*x-1)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3/((-a^2*c*x^2 + c)^(7/2)*(-a^2*x^2 + 1)^(3/2)), x)

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Fricas [A]  time = 2.93026, size = 1175, normalized size = 4.23 \begin{align*} \left [\frac{15 \,{\left (a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) + 4 \,{\left (32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{384 \,{\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac{15 \,{\left (a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right ) + 2 \,{\left (32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \,{\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

[1/384*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(a^6*c*x
^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x
^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(32*a^5*x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*
x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*
c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/192*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3
*a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(32*a^5*
x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - 3
*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]

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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)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)^3/((-a^2*c*x^2 + c)^(7/2)*(-a^2*x^2 + 1)^(3/2)), x)