3.669 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^4} \, dx\)

Optimal. Leaf size=163 \[ \frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4} \]

[Out]

(a^6*x^7*(1 - a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) - (a^4*x^5*(7 - 8*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*x^
3*(35 - 48*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) - (x*(35 - 64*a*x))/(35*c^4*Sqrt[1 - a^2*x^2]) + (128*Sqrt[1 -
a^2*x^2])/(35*a*c^4) + ArcSin[a*x]/(a*c^4)

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Rubi [A]  time = 0.240775, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6157, 6149, 819, 641, 216} \[ \frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4),x]

[Out]

(a^6*x^7*(1 - a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) - (a^4*x^5*(7 - 8*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*x^
3*(35 - 48*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) - (x*(35 - 64*a*x))/(35*c^4*Sqrt[1 - a^2*x^2]) + (128*Sqrt[1 -
a^2*x^2])/(35*a*c^4) + ArcSin[a*x]/(a*c^4)

Rule 6157

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

Rule 6149

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=\frac{a^8 \int \frac{e^{-\tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac{a^8 \int \frac{x^8 (1-a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^6 \int \frac{x^6 (7-8 a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 \int \frac{x^4 (35-48 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 \int \frac{x^2 (105-192 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105-384 a x}{\sqrt{1-a^2 x^2}} \, dx}{105 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.0961641, size = 126, normalized size = 0.77 \[ \frac{-105 a^7 x^7-281 a^6 x^6+559 a^5 x^5+965 a^4 x^4-715 a^3 x^3-1065 a^2 x^2+105 (a x-1)^2 (a x+1)^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+279 a x+384}{105 a c^4 (a x-1)^2 (a x+1)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4),x]

[Out]

(384 + 279*a*x - 1065*a^2*x^2 - 715*a^3*x^3 + 965*a^4*x^4 + 559*a^5*x^5 - 281*a^6*x^6 - 105*a^7*x^7 + 105*(-1
+ a*x)^2*(1 + a*x)^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(105*a*c^4*(-1 + a*x)^2*(1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.069, size = 438, normalized size = 2.7 \begin{align*}{\frac{1}{14\,{a}^{5}{c}^{4} \left ( x+{a}^{-1} \right ) ^{4}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{187}{672\,{a}^{4}{c}^{4} \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{53}{960\,{a}^{4}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{47}{128\,{a}^{3}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{187}{256\,a{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}-{\frac{187}{256\,{c}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{160\,{a}^{5}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-4}}+{\frac{35}{32\,{a}^{3}{c}^{4} \left ( x+{a}^{-1} \right ) ^{2}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{443}{256\,a{c}^{4}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{443}{256\,{c}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{112\,{a}^{6}{c}^{4} \left ( x+{a}^{-1} \right ) ^{5}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x)

[Out]

1/14/a^5/c^4/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-187/672/a^4/c^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a
))^(3/2)+53/960/a^4/c^4/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)+47/128/a^3/c^4/(x-1/a)^2*(-a^2*(x-1/a)^2-
2*a*(x-1/a))^(3/2)+187/256/a/c^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-187/256/c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)
*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))+1/160/a^5/c^4/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)+35/32/a^3/c^
4/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+443/256/a/c^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+443/256/c^4/(a
^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))-1/112/a^6/c^4/(x+1/a)^5*(-a^2*(x+1/a)^2+2*a
*(x+1/a))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a^2*x^2))^4), x)

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Fricas [A]  time = 2.38672, size = 632, normalized size = 3.88 \begin{align*} \frac{384 \, a^{7} x^{7} + 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} - 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} + 1152 \, a^{2} x^{2} - 384 \, a x - 210 \,{\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 )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right )} \sqrt{-a^{2} x^{2} + 1} - 384}{105 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

1/105*(384*a^7*x^7 + 384*a^6*x^6 - 1152*a^5*x^5 - 1152*a^4*x^4 + 1152*a^3*x^3 + 1152*a^2*x^2 - 384*a*x - 210*(
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)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(
a*x)) + (105*a^7*x^7 + 281*a^6*x^6 - 559*a^5*x^5 - 965*a^4*x^4 + 715*a^3*x^3 + 1065*a^2*x^2 - 279*a*x - 384)*s
qrt(-a^2*x^2 + 1) - 384)/(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*} \frac{a^{8} \int \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**4,x)

[Out]

a**8*Integral(x**8*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + a**8*x**8 - 4*a**7*x**7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a
**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x + 1), x)/c**4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a^2*x^2))^4), x)