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]

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

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Rubi [A]  time = 0.56, antiderivative size = 185, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [A]  time = 0.15, 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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fricas [A]  time = 0.72, size = 281, normalized size = 1.52 \[ \frac {1664 \, a^{7} x^{7} - 4992 \, a^{6} x^{6} + 1664 \, a^{5} x^{5} + 8320 \, a^{4} x^{4} - 8320 \, a^{3} x^{3} - 1664 \, a^{2} x^{2} + 4992 \, a x + 1890 \, {\left (a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (315 \, a^{7} x^{7} - 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} + 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} + 339 \, a^{2} x^{2} + 4047 \, a x - 1664\right )} \sqrt {-a^{2} x^{2} + 1} - 1664}{315 \, {\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]

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)

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maple [B]  time = 0.06, size = 367, normalized size = 1.98 \[ -\frac {a \,x^{2}}{c^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {9}{a \,c^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {16 x}{c^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} \sqrt {a^{2}}}+\frac {5111}{2520 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {26633}{5040 a^{2} c^{4} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {26633 x}{2520 c^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {1}{18 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {125}{252 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {1}{48 a^{2} c^{4} \left (x +\frac {1}{a}\right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}+\frac {x}{24 c^{4} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}} \]

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))+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)+125/252/a^4/c^4/(x-1/a)^3/(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]

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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mupad [B]  time = 4.43, size = 2165, normalized size = 11.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - a^2*x^2)^(1/2)/(72*(-a^2)^(1/2)*(c^4*1i + 5*c^4*x*(-a^2)^(1/2) + a^2*c^4*x^2*10i + a^4*c^4*x^4*5i + 10*a^
2*c^4*x^3*(-a^2)^(1/2) + a^4*c^4*x^5*(-a^2)^(1/2))) - (109*a^9*(1 - a^2*x^2)^(1/2))/(1344*(a^10*c^4 + c^4*x*(-
a^2)^(11/2)*4i + 6*a^12*c^4*x^2 + a^14*c^4*x^4 + a^2*c^4*x^3*(-a^2)^(11/2)*4i)) - (145*a^11*(1 - a^2*x^2)^(1/2
))/(4032*(a^12*c^4 - c^4*x*(-a^2)^(13/2)*4i + 6*a^14*c^4*x^2 + a^16*c^4*x^4 - a^2*c^4*x^3*(-a^2)^(13/2)*4i)) -
(145*a^11*(1 - a^2*x^2)^(1/2))/(4032*(a^12*c^4 + c^4*x*(-a^2)^(13/2)*4i + 6*a^14*c^4*x^2 + a^16*c^4*x^4 + a^2
*c^4*x^3*(-a^2)^(13/2)*4i)) - (14711*a^9*(1 - a^2*x^2)^(1/2))/(26880*(a^10*c^4 - c^4*x*(-a^2)^(11/2)*2i + a^12
*c^4*x^2)) - (14711*a^9*(1 - a^2*x^2)^(1/2))/(26880*(a^10*c^4 + c^4*x*(-a^2)^(11/2)*2i + a^12*c^4*x^2)) - (894
7*a^11*(1 - a^2*x^2)^(1/2))/(16128*(a^12*c^4 - c^4*x*(-a^2)^(13/2)*2i + a^14*c^4*x^2)) - (8947*a^11*(1 - a^2*x
^2)^(1/2))/(16128*(a^12*c^4 + c^4*x*(-a^2)^(13/2)*2i + a^14*c^4*x^2)) - (3*asinh(x*(-a^2)^(1/2)))/(c^4*(-a^2)^
(1/2)) - (109*a^9*(1 - a^2*x^2)^(1/2))/(1344*(a^10*c^4 - c^4*x*(-a^2)^(11/2)*4i + 6*a^12*c^4*x^2 + a^14*c^4*x^
4 - a^2*c^4*x^3*(-a^2)^(11/2)*4i)) - (1 - a^2*x^2)^(1/2)/(72*(-a^2)^(1/2)*(c^4*1i - 5*c^4*x*(-a^2)^(1/2) + a^2
*c^4*x^2*10i + a^4*c^4*x^4*5i - 10*a^2*c^4*x^3*(-a^2)^(1/2) - a^4*c^4*x^5*(-a^2)^(1/2))) + (a^9*(1 - a^2*x^2)^
(1/2)*1i)/(96*(a^10*c^4*1i + 5*c^4*x*(-a^2)^(11/2) + a^12*c^4*x^2*10i + a^14*c^4*x^4*5i + 10*a^2*c^4*x^3*(-a^2
)^(11/2) + a^4*c^4*x^5*(-a^2)^(11/2))) + (a^9*(1 - a^2*x^2)^(1/2)*1i)/(96*(a^10*c^4*1i - 5*c^4*x*(-a^2)^(11/2)
+ a^12*c^4*x^2*10i + a^14*c^4*x^4*5i - 10*a^2*c^4*x^3*(-a^2)^(11/2) - a^4*c^4*x^5*(-a^2)^(11/2))) + (a^11*(1
- a^2*x^2)^(1/2)*1i)/(288*(a^12*c^4*1i + 5*c^4*x*(-a^2)^(13/2) + a^14*c^4*x^2*10i + a^16*c^4*x^4*5i + 10*a^2*c
^4*x^3*(-a^2)^(13/2) + a^4*c^4*x^5*(-a^2)^(13/2))) + (a^11*(1 - a^2*x^2)^(1/2)*1i)/(288*(a^12*c^4*1i - 5*c^4*x
*(-a^2)^(13/2) + a^14*c^4*x^2*10i + a^16*c^4*x^4*5i - 10*a^2*c^4*x^3*(-a^2)^(13/2) - a^4*c^4*x^5*(-a^2)^(13/2)
)) + (1507*(1 - a^2*x^2)^(1/2))/(3360*(-a^2)^(1/2)*(c^4*1i + 3*c^4*x*(-a^2)^(1/2) + a^2*c^4*x^2*3i + a^2*c^4*x
^3*(-a^2)^(1/2))) - (1507*(1 - a^2*x^2)^(1/2))/(3360*(-a^2)^(1/2)*(c^4*1i - 3*c^4*x*(-a^2)^(1/2) + a^2*c^4*x^2
*3i - a^2*c^4*x^3*(-a^2)^(1/2))) + (a^9*(1 - a^2*x^2)^(1/2)*1231i)/(4480*(a^10*c^4*1i + 3*c^4*x*(-a^2)^(11/2)
+ a^12*c^4*x^2*3i + a^2*c^4*x^3*(-a^2)^(11/2))) + (a^9*(1 - a^2*x^2)^(1/2)*1231i)/(4480*(a^10*c^4*1i - 3*c^4*x
*(-a^2)^(11/2) + a^12*c^4*x^2*3i - a^2*c^4*x^3*(-a^2)^(11/2))) + (a^11*(1 - a^2*x^2)^(1/2)*467i)/(2688*(a^12*c
^4*1i + 3*c^4*x*(-a^2)^(13/2) + a^14*c^4*x^2*3i + a^2*c^4*x^3*(-a^2)^(13/2))) + (a^11*(1 - a^2*x^2)^(1/2)*467i
)/(2688*(a^12*c^4*1i - 3*c^4*x*(-a^2)^(13/2) + a^14*c^4*x^2*3i - a^2*c^4*x^3*(-a^2)^(13/2))) + (862*(1 - a^2*x
^2)^(1/2))/(315*(c^4*1i + c^4*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (862*(1 - a^2*x^2)^(1/2))/(315*(c^4*1i - c^4*x*(
-a^2)^(1/2))*(-a^2)^(1/2)) + (a^9*(1 - a^2*x^2)^(1/2)*25609i)/(26880*(a^10*c^4*1i + c^4*x*(-a^2)^(11/2))) + (a
^9*(1 - a^2*x^2)^(1/2)*25609i)/(26880*(a^10*c^4*1i - c^4*x*(-a^2)^(11/2))) + (a^11*(1 - a^2*x^2)^(1/2)*31373i)
/(16128*(a^12*c^4*1i + c^4*x*(-a^2)^(13/2))) + (a^11*(1 - a^2*x^2)^(1/2)*31373i)/(16128*(a^12*c^4*1i - c^4*x*(
-a^2)^(13/2))) + (1 - a^2*x^2)^(1/2)/(a*c^4) + ((1 - a^2*x^2)^(1/2)*59i)/(504*(-a^2)^(1/2)*(c^4 - c^4*x*(-a^2)
^(1/2)*4i + 6*a^2*c^4*x^2 + a^4*c^4*x^4 - a^2*c^4*x^3*(-a^2)^(1/2)*4i)) - ((1 - a^2*x^2)^(1/2)*59i)/(504*(-a^2
)^(1/2)*(c^4 + c^4*x*(-a^2)^(1/2)*4i + 6*a^2*c^4*x^2 + a^4*c^4*x^4 + a^2*c^4*x^3*(-a^2)^(1/2)*4i)) + ((1 - a^2
*x^2)^(1/2)*22007i)/(20160*(-a^2)^(1/2)*(c^4 - c^4*x*(-a^2)^(1/2)*2i + a^2*c^4*x^2)) - ((1 - a^2*x^2)^(1/2)*22
007i)/(20160*(-a^2)^(1/2)*(c^4 + c^4*x*(-a^2)^(1/2)*2i + a^2*c^4*x^2))

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

[Out]

Timed out

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