3.7.85 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^4} \, dx\) [685]

Optimal. Leaf size=189 \[ -\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \text {ArcSin}(a x)}{a c^4} \]

[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 = 0.40, antiderivative size = 189, 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 = {6292, 6284, 1649, 1828, 655, 222} \begin {gather*} -\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 {\sqrt {1-a^2 x^2}}{a c^4}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 \text {ArcSin}(a x)}{a c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4),x]

[Out]

-1/9*(1 - a*x)^3/(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 222

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 655

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 1649

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 1828

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 6284

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 6292

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u/x^(2*p))*(1
- a^2*x^2)^p*E^(n*ArcTanh[a*x]), 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 = 0.11, size = 122, normalized size = 0.65 \begin {gather*} \frac {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) (1+a x)^4 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{315 a (-1+a x) (c+a c x)^4 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(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)*(1 + a*x)^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(315*a*(-1 + a*x)*(c + a*c*x)^4*Sqrt[1 - a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1699\) vs. \(2(167)=334\).
time = 0.85, size = 1700, normalized size = 8.99

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{4}}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{8} \sqrt {a^{2}}}-\frac {59 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{252 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {1507 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{1680 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {691 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{315 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {113591 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{20160 a^{10} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{96 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {31 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{192 a^{10} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{36 a^{14} \left (x +\frac {1}{a}\right )^{5}}\right ) a^{8}}{c^{4}}\)	\(326\)
default	\(\text {Expression too large to display}\)	\(1700\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8/c^4*(7/128/a^10*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(
3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/
(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))+9/256/a^11*(1/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+2*a*(-1/a/
(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/
a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a
))^(1/2))))))-7/256/a^9*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)
^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))+163/256/a^11*
(-1/a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(
1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2
/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))))-3/8/a^14*(-1/7/a/(x+1/a)^6*(-a^2*(x+
1/a)^2+2*a*(x+1/a))^(5/2)-1/35/(x+1/a)^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2))-7/64/a^10*(1/a/(x+1/a)^2*(-a^2*(x
+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*
(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))+1/128
/a^12*(1/3/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+1/3*a*(1/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5
/2)+2*a*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4
*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a
)^2-2*a*(x-1/a))^(1/2)))))))-29/160/a^14/(x+1/a)^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+1/16/a^15*(-1/9/a/(x+1/a
)^7*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+2/9*a*(-1/7/a/(x+1/a)^6*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-1/35/(x+1/a)
^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)))-35/32/a^12*(-1/3/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-1/3*a*
(-1/a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(
1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2
/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))))+7/256/a^9*(1/3*(-a^2*(x-1/a)^2-2*a*
(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2
)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 278, normalized size = 1.47 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 - 1
890*(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{8} \left (\int \frac {x^{8} \sqrt {- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{2} x^{10} \sqrt {- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\right )\, dx\right )}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*(Integral(x**8*sqrt(-a**2*x**2 + 1)/(a**11*x**11 + 3*a**10*x**10 - a**9*x**9 - 11*a**8*x**8 - 6*a**7*x**7
+ 14*a**6*x**6 + 14*a**5*x**5 - 6*a**4*x**4 - 11*a**3*x**3 - a**2*x**2 + 3*a*x + 1), x) + Integral(-a**2*x**1
0*sqrt(-a**2*x**2 + 1)/(a**11*x**11 + 3*a**10*x**10 - a**9*x**9 - 11*a**8*x**8 - 6*a**7*x**7 + 14*a**6*x**6 +
14*a**5*x**5 - 6*a**4*x**4 - 11*a**3*x**3 - a**2*x**2 + 3*a*x + 1), x))/c**4

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.79, size = 671, normalized size = 3.55 \begin {gather*} \frac {a\,\sqrt {1-a^2\,x^2}}{96\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {67\,a\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^4\,x^2+2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {\sqrt {1-a^2\,x^2}}{36\,\sqrt {-a^2}\,\left (5\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+10\,a^2\,c^4\,x^3\,\sqrt {-a^2}+5\,a^3\,c^4\,x^4\,\sqrt {-a^2}+a^4\,c^4\,x^5\,\sqrt {-a^2}+10\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{10\,\left (a^6\,c^4\,x^2+2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {1759\,a^8\,\sqrt {1-a^2\,x^2}}{2520\,\left (a^{11}\,c^4\,x^2+2\,a^{10}\,c^4\,x+a^9\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {a\,\sqrt {1-a^2\,x^2}}{4\,\left (a^6\,c^4\,x^4+4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2+4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {113591\,\sqrt {1-a^2\,x^2}}{20160\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {31\,\sqrt {1-a^2\,x^2}}{192\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {1507\,\sqrt {1-a^2\,x^2}}{1680\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}+3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {a^{10}\,\sqrt {1-a^2\,x^2}}{63\,\left (a^{15}\,c^4\,x^4+4\,a^{14}\,c^4\,x^3+6\,a^{13}\,c^4\,x^2+4\,a^{12}\,c^4\,x+a^{11}\,c^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*(1 - a^2*x^2)^(1/2))/(96*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) + (67*a*(1 - a^2*x^2)^(1/2))/(24*(a^2*c^4 +
2*a^3*c^4*x + a^4*c^4*x^2)) + (1 - a^2*x^2)^(1/2)/(36*(-a^2)^(1/2)*(5*c^4*x*(-a^2)^(1/2) + (c^4*(-a^2)^(1/2))
/a + 10*a^2*c^4*x^3*(-a^2)^(1/2) + 5*a^3*c^4*x^4*(-a^2)^(1/2) + a^4*c^4*x^5*(-a^2)^(1/2) + 10*a*c^4*x^2*(-a^2)
^(1/2))) - (3*asinh(x*(-a^2)^(1/2)))/(c^4*(-a^2)^(1/2)) + (a^3*(1 - a^2*x^2)^(1/2))/(10*(a^4*c^4 + 2*a^5*c^4*x
+ a^6*c^4*x^2)) - (1759*a^8*(1 - a^2*x^2)^(1/2))/(2520*(a^9*c^4 + 2*a^10*c^4*x + a^11*c^4*x^2)) - (1 - a^2*x^
2)^(1/2)/(a*c^4) + (a*(1 - a^2*x^2)^(1/2))/(4*(a^2*c^4 + 4*a^3*c^4*x + 6*a^4*c^4*x^2 + 4*a^5*c^4*x^3 + a^6*c^4
*x^4)) + (113591*(1 - a^2*x^2)^(1/2))/(20160*(-a^2)^(1/2)*(c^4*x*(-a^2)^(1/2) + (c^4*(-a^2)^(1/2))/a)) - (31*(
1 - a^2*x^2)^(1/2))/(192*(-a^2)^(1/2)*(c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) + (1507*(1 - a^2*x^2)^(1/2)
)/(1680*(-a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) + (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3*(-a^2)^(1/2) + 3*a*c^4*x^2*(-a
^2)^(1/2))) - (a^10*(1 - a^2*x^2)^(1/2))/(63*(a^11*c^4 + 4*a^12*c^4*x + 6*a^13*c^4*x^2 + 4*a^14*c^4*x^3 + a^15
*c^4*x^4))

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