3.7.52 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^4} \, dx\) [652]

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

[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 = 0.37, 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 = {6292, 6283, 1649, 1828, 655, 222} \begin {gather*} \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 \text {ArcSin}(a x)}{a c^4} \end {gather*}

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 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 6283

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 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 = 0.10, size = 124, normalized size = 0.67 \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)^4 (1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{315 a c^4 (-1+a x)^4 (1+a x) \sqrt {1-a^2 x^2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(163)=326\).
time = 0.80, size = 872, normalized size = 4.71

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{4}}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{8} \sqrt {a^{2}}}+\frac {1507 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{1680 a^{12} \left (x -\frac {1}{a}\right )^{3}}+\frac {691 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{315 a^{11} \left (x -\frac {1}{a}\right )^{2}}+\frac {113591 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{20160 a^{10} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{36 a^{14} \left (x -\frac {1}{a}\right )^{5}}+\frac {59 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{252 a^{13} \left (x -\frac {1}{a}\right )^{4}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{96 a^{11} \left (x +\frac {1}{a}\right )^{2}}-\frac {31 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{192 a^{10} \left (x +\frac {1}{a}\right )}\right ) a^{8}}{c^{4}}\)	\(345\)
default	\(\frac {a^{8} \left (\frac {-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{5}}+\frac {\frac {3 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a^{6}}+\frac {7}{a^{9} \sqrt {-a^{2} x^{2}+1}}+\frac {13 x}{a^{8} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {97}{40 a \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 {291 a \left (\frac {1}{3 a \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 {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{40}}{a^{10}}+\frac {\frac {15}{28 a \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 {15 a \left (\frac {1}{5 a \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 {3 a \left (\frac {1}{3 a \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 {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{11}}+\frac {-\frac {1}{3 a \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 {-2 a^{2} \left (x +\frac {1}{a}\right )+2 a}{3 a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}}{16 a^{9}}+\frac {\frac {1}{9 a \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 {5 a \left (\frac {1}{7 a \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 {4 a \left (\frac {1}{5 a \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 {3 a \left (\frac {1}{3 a \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 {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}\right )}{9}}{2 a^{12}}+\frac {\frac {117}{16 a \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 {117 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{16 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{9}}\right )}{c^{4}}\)	\(872\)

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,method=_RETURNVERBOSE)

[Out]

a^8/c^4*(1/a^5*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+3/a^6*(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a
^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))+7/a^9/(-a^2*x^2+1)^(1/2)+13/a^8*x/(-a^2*x^2+1)^(1/2)+97/8/
a^10*(1/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/
2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))+15/4/a^11*(1/7/a/(x-1/a)^3/(-a^2*(x-1/a)^2-
2*a*(x-1/a))^(1/2)-4/7*a*(1/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a
)^2-2*a*(x-1/a))^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))+1/16/a^9*(-1/3/a/(x+1/
a)/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/a*(-2*a^2*(x+1/a)+2*a)/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))+1/2/a^12*
(1/9/a/(x-1/a)^4/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-5/9*a*(1/7/a/(x-1/a)^3/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-
4/7*a*(1/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1
/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))+351/16/a^9*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^
2-2*a*(x-1/a))^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-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((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 = 0.46, size = 281, normalized size = 1.52 \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((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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{8} \int \frac {x^{8}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 3 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \end {gather*}

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]

a**8*Integral(x**8/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 3*a**6*x**6*sqrt(-a**2*x**2 + 1) - a**5*x**5*sqrt(-a**2*
x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 +
1) - 3*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 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((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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Mupad [B]
time = 4.43, size = 2165, normalized size = 11.70 \begin {gather*} \text {Too large to display} \end {gather*}

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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