3.652 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^4} \, dx\)

Optimal. Leaf size=185 \[ \frac{(a x+1)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (a x+1)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (a x+1)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{a c^4}+\frac{4 (431 a x+630)}{315 a c^4 \sqrt{1-a^2 x^2}}-\frac{2 (829 a x+1155)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \sin ^{-1}(a x)}{a c^4} \]

[Out]

(1 + a*x)^3/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (22*(1 + a*x)^2)/(21*a*c^4*(1 - a^2*x^2)^(7/2)) + (478*(1 + a*x))/
(105*a*c^4*(1 - a^2*x^2)^(5/2)) - (2*(1155 + 829*a*x))/(315*a*c^4*(1 - a^2*x^2)^(3/2)) + (4*(630 + 431*a*x))/(
315*a*c^4*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^4) - (3*ArcSin[a*x])/(a*c^4)

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Rubi [A]  time = 0.564101, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6157, 6148, 1635, 1814, 641, 216} \[ \frac{(a x+1)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (a x+1)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (a x+1)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{a c^4}+\frac{4 (431 a x+630)}{315 a c^4 \sqrt{1-a^2 x^2}}-\frac{2 (829 a x+1155)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \sin ^{-1}(a x)}{a c^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^4,x]

[Out]

(1 + a*x)^3/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (22*(1 + a*x)^2)/(21*a*c^4*(1 - a^2*x^2)^(7/2)) + (478*(1 + a*x))/
(105*a*c^4*(1 - a^2*x^2)^(5/2)) - (2*(1155 + 829*a*x))/(315*a*c^4*(1 - a^2*x^2)^(3/2)) + (4*(630 + 431*a*x))/(
315*a*c^4*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^4) - (3*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 6148

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] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

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^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=\frac{a^8 \int \frac{e^{3 \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)^3}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{a^8 \int \frac{(1+a x)^2 \left (\frac{3}{a^8}+\frac{9 x}{a^7}+\frac{9 x^2}{a^6}+\frac{9 x^3}{a^5}+\frac{9 x^4}{a^4}+\frac{9 x^5}{a^3}+\frac{9 x^6}{a^2}+\frac{9 x^7}{a}\right )}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{a^8 \int \frac{(1+a x) \left (\frac{111}{a^8}+\frac{378 x}{a^7}+\frac{315 x^2}{a^6}+\frac{252 x^3}{a^5}+\frac{189 x^4}{a^4}+\frac{126 x^5}{a^3}+\frac{63 x^6}{a^2}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{63 c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (1+a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{a^8 \int \frac{\frac{879}{a^8}+\frac{4725 x}{a^7}+\frac{3150 x^2}{a^6}+\frac{1890 x^3}{a^5}+\frac{945 x^4}{a^4}+\frac{315 x^5}{a^3}}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{315 c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (1+a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1155+829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^8 \int \frac{\frac{2337}{a^8}+\frac{6615 x}{a^7}+\frac{2835 x^2}{a^6}+\frac{945 x^3}{a^5}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{945 c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (1+a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1155+829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (630+431 a x)}{315 a c^4 \sqrt{1-a^2 x^2}}-\frac{a^8 \int \frac{\frac{2835}{a^8}+\frac{945 x}{a^7}}{\sqrt{1-a^2 x^2}} \, dx}{945 c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (1+a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1155+829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (630+431 a x)}{315 a c^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^4}-\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{(1+a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{22 (1+a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{478 (1+a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1155+829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (630+431 a x)}{315 a c^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^4}-\frac{3 \sin ^{-1}(a x)}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.146165, size = 124, normalized size = 0.67 \[ \frac{-315 a^7 x^7+2669 a^6 x^6-2967 a^5 x^5-4029 a^4 x^4+7399 a^3 x^3-339 a^2 x^2-945 (a x-1)^4 (a x+1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-4047 a x+1664}{315 a c^4 (a x-1)^4 (a x+1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^4,x]

[Out]

(1664 - 4047*a*x - 339*a^2*x^2 + 7399*a^3*x^3 - 4029*a^4*x^4 - 2967*a^5*x^5 + 2669*a^6*x^6 - 315*a^7*x^7 - 945
*(-1 + a*x)^4*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(315*a*c^4*(-1 + a*x)^4*(1 + a*x)*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.061, size = 367, normalized size = 2. \begin{align*} -{\frac{a{x}^{2}}{{c}^{4}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+9\,{\frac{1}{a{c}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}}+16\,{\frac{x}{{c}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}}-3\,{\frac{1}{{c}^{4}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{125}{252\,{a}^{4}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-3}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{5111}{2520\,{a}^{3}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-2}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{26633}{5040\,{a}^{2}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}-{\frac{26633\,x}{2520\,{c}^{4}}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{1}{18\,{a}^{5}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-4}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}-{\frac{1}{48\,{a}^{2}{c}^{4} \left ( x+{a}^{-1} \right ) }{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}}+{\frac{x}{24\,{c}^{4}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-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)/(c-c/a^2/x^2)^4,x)

[Out]

-a/c^4*x^2/(-a^2*x^2+1)^(1/2)+9/a/c^4/(-a^2*x^2+1)^(1/2)+16*x/c^4/(-a^2*x^2+1)^(1/2)-3/c^4/(a^2)^(1/2)*arctan(
(a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+125/252/a^4/c^4/(x-1/a)^3/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+5111/2520/a^3/c
^4/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+26633/5040/a^2/c^4/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-
26633/2520/c^4/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x+1/18/a^5/c^4/(x-1/a)^4/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-
1/48/a^2/c^4/(x+1/a)/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/24/c^4/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x

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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} x^{2} + 1\right )}^{\frac{3}{2}}{\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((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^4), x)

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Fricas [A]  time = 2.58015, size = 645, normalized size = 3.49 \begin{align*} \frac{1664 \, a^{7} x^{7} - 4992 \, a^{6} x^{6} + 1664 \, a^{5} x^{5} + 8320 \, a^{4} x^{4} - 8320 \, a^{3} x^{3} - 1664 \, a^{2} x^{2} + 4992 \, a x + 1890 \,{\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 )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (315 \, a^{7} x^{7} - 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} + 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} + 339 \, a^{2} x^{2} + 4047 \, a x - 1664\right )} \sqrt{-a^{2} x^{2} + 1} - 1664}{315 \,{\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 )}} \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)/(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

1/315*(1664*a^7*x^7 - 4992*a^6*x^6 + 1664*a^5*x^5 + 8320*a^4*x^4 - 8320*a^3*x^3 - 1664*a^2*x^2 + 4992*a*x + 18
90*(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)*arctan((sqrt(-a^2*x^2 + 1) -
1)/(a*x)) + (315*a^7*x^7 - 2669*a^6*x^6 + 2967*a^5*x^5 + 4029*a^4*x^4 - 7399*a^3*x^3 + 339*a^2*x^2 + 4047*a*x
- 1664)*sqrt(-a^2*x^2 + 1) - 1664)/(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)/(c-c/a**2/x**2)**4,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} x^{2} + 1\right )}^{\frac{3}{2}}{\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((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^4), x)