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

Optimal. Leaf size=277 \[ \frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}} \]

[Out]

Sqrt[1 - a^2*x^2]/(24*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*Sqr
t[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(16*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)^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]) + (5*Sqrt[1 -
a^2*x^2]*ArcTanh[a*x])/(16*a*c^3*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.132034, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6143, 6140, 44, 207} \[ \frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (1-a x)^3 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - a^2*x^2]/(24*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*Sqr
t[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(16*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)^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]) + (5*Sqrt[1 -
a^2*x^2]*ArcTanh[a*x])/(16*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^{\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^{\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)^4 (1+a x)^3} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{8 (-1+a x)^4}-\frac{3}{16 (-1+a x)^3}+\frac{3}{16 (-1+a x)^2}+\frac{1}{16 (1+a x)^3}+\frac{1}{8 (1+a x)^2}-\frac{5}{16 \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}}{24 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{3 \sqrt{1-a^2 x^2}}{16 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)^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{\left (5 \sqrt{1-a^2 x^2}\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{16 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{24 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{3 \sqrt{1-a^2 x^2}}{16 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)^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{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0789628, size = 103, normalized size = 0.37 \[ \frac{\sqrt{1-a^2 x^2} \left (-15 a^4 x^4+15 a^3 x^3+25 a^2 x^2-25 a x+15 (a x-1)^3 (a x+1)^2 \tanh ^{-1}(a x)-8\right )}{48 a c^3 (a x-1)^3 (a x+1)^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(-8 - 25*a*x + 25*a^2*x^2 + 15*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^3*(1 + a*x)^2*ArcTanh[a
*x]))/(48*a*c^3*(-1 + a*x)^3*(1 + a*x)^2*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.103, 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}-15\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+15\,\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}+30\,{x}^{3}{a}^{3}+30\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-30\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+50\,{a}^{2}{x}^{2}+15\,ax\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) xa-50\,ax-15\,\ln \left ( ax+1 \right ) +15\,\ln \left ( ax-1 \right ) -16}{ \left ( 96\,{a}^{2}{x}^{2}-96 \right ){c}^{4}a \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{3}}\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)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(7/2),x)

[Out]

-1/96*(-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^5-15*ln(a*x+1)*a^4*x^
4+15*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+30*x^3*a^3+30*ln(a*x+1)*a^2*x^2-30
*ln(a*x-1)*a^2*x^2+50*a^2*x^2+15*a*x*ln(a*x+1)-15*ln(a*x-1)*x*a-50*a*x-15*ln(a*x+1)+15*ln(a*x-1)-16)/(a^2*x^2-
1)/c^4/a/(a*x+1)^2/(a*x-1)^3

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

[Out]

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

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Fricas [A]  time = 2.13687, size = 1165, normalized size = 4.21 \begin{align*} \left [\frac{15 \,{\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 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 (8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \,{\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}, \frac{15 \,{\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 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 (8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \,{\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - 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)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

[1/192*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - 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*(8*a^5*x^5 + 7*a^4*x^4 - 31*a^3*x^3 - 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2
+ c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c
^4*x^2 - a^2*c^4*x + a*c^4), 1/96*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - 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*(8*a^5*x^5 +
7*a^4*x^4 - 31*a^3*x^3 - 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - a^7*c^4*
x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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