3.24.98 \(\int \frac {-1+a k x+k x^2}{(1+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\) [2398]

Optimal. Leaf size=193 \[ \frac {\text {ArcTan}\left (\frac {\left (-1-2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}+\frac {\text {ArcTan}\left (\frac {\left (-1+2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}-\frac {a \sqrt {k} \tanh ^{-1}\left (\frac {\left (2 \sqrt {k}+2 k^{3/2}\right ) x^2}{1+2 k x^2+k^2 x^4+\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 (1+k)} \]

[Out]

arctan((-1-2*k^(1/2)-k)*x/(-1+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(1+k)+arctan((-1+2*k^(1/2)-k)*x/(-1+k*x^2
+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(1+k)-a*k^(1/2)*arctanh((2*k^(1/2)+2*k^(3/2))*x^2/(1+2*k*x^2+k^2*x^4+(k*x^2-
1)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(2+2*k)

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Rubi [A]
time = 0.23, antiderivative size = 98, normalized size of antiderivative = 0.51, number of steps used = 8, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {1976, 1701, 1712, 209, 12, 1261, 738, 212} \begin {gather*} -\frac {a \sqrt {k} \tanh ^{-1}\left (\frac {-k (k+1) x^2+k+1}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (k+1)}-\frac {\text {ArcTan}\left (\frac {(k+1) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{k+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + a*k*x + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]

[Out]

-(ArcTan[((1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/(1 + k)) - (a*Sqrt[k]*ArcTanh[(1 + k - k*(1 + k)*x^2)
/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(2*(1 + k))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

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

Rule 212

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1701

Int[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{r = Expo
n[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] + Int[x*
Sum[Coeff[Pr, x, 2*k + 1]*x^(2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b,
c, d, e, p, q}, x] && PolyQ[Pr, x] &&  !PolyQ[Pr, x^2]

Rule 1712

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+a k x+k x^2}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {2-a k x}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {2-a k x}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {2-\frac {a k}{\sqrt {-k}}}{2 \left (1-\sqrt {-k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}+\frac {2+\frac {a k}{\sqrt {-k}}}{2 \left (1+\sqrt {-k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2-a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (1+\sqrt {-k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2+a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (1-\sqrt {-k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2-a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2+a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\left (2-a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2+a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2-a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2+a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\left (2-a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+k x) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{4 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2+a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+k x) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{4 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2-a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2+a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\left (2-a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-k-\left (-k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\left (2+a \sqrt {-k}\right ) \sqrt {-k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-k-\left (-k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (2 \sqrt {-k}-a k\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 \sqrt {k} (1+k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {-k}+a k\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 \sqrt {k} (1+k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2-a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2+a \sqrt {-k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.17, size = 158, normalized size = 0.82 \begin {gather*} \frac {a \sqrt {k} \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )+(1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\text {ArcSin}(x)\left |k^2\right .\right )-2 (1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\text {ArcSin}(x)\left |k^2\right .\right )}{(1+k) \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + a*k*x + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]

[Out]

(a*Sqrt[k]*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*ArcTanh[Sqrt[-1 + k^2*x^2]/(Sqrt[k]*Sqrt[-1 + x^2])] + (1 + k)*Sq
rt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - 2*(1 + k)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi
[-k, ArcSin[x], k^2])/((1 + k)*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.11, size = 337, normalized size = 1.75

method	result	size
elliptic	\(-\frac {a \ln \left (\frac {\frac {2 k^{2}+4 k +2}{k}+\left (-k^{2}-2 k -1\right ) \left (x^{2}+\frac {1}{k}\right )+2 \sqrt {\frac {k^{2}+2 k +1}{k}}\, \sqrt {k^{2} \left (x^{2}+\frac {1}{k}\right )^{2}+\left (-k^{2}-2 k -1\right ) \left (x^{2}+\frac {1}{k}\right )+\frac {k^{2}+2 k +1}{k}}}{x^{2}+\frac {1}{k}}\right )}{2 \sqrt {\frac {k^{2}+2 k +1}{k}}}+\frac {\arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (1+k \right )}\right )}{1+k}\)	\(160\)
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {a \arctanh \left (\frac {k \,x^{2}}{\sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {k}{2 \sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {k^{2} x^{2}}{2 \sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {1}{2 \sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}\, k}+\frac {x^{2}}{2 \sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {1}{\sqrt {\frac {1}{k}+2+k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )}{2 \sqrt {\frac {1}{k}+2+k}}-\frac {2 \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticPi \left (x , -k , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x^2+1)^(1/2)*(-k^2*x^2+1)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*EllipticF(x,k)+1/2*a/(1/k+2+k)^(1/2)*arctanh(1
/(1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*k*x^2-1/2/(1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*k+1/2/(
1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*k^2*x^2-1/2/(1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)/k+1/2/(
1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*x^2-1/(1/k+2+k)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2))-2*(-x^2+1)^(
1/2)*(-k^2*x^2+1)^(1/2)/(k^2*x^4-k^2*x^2-x^2+1)^(1/2)*EllipticPi(x,-k,k)

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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*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (167) = 334\).
time = 1.25, size = 1809, normalized size = 9.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*
(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 + 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k
+ 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) + (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 +
2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x + (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 +
6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4
*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3
+ 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) + 1/8*sqrt((a^2*k - 4*sqrt
(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)
*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 + 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 +
4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) - (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a
^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x + (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)
*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sq
rt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2
+ 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) - 1/8*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*
k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k
^2 + 4*a*k)*x^2 - 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))
*x + 4*a) + (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*
a*k + a)*x - (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2
*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 +
6*k^2 + 4*k + 1)))*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*
k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) + 1/8*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*
k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 - 2*(a^2*
k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) - (2*a^2*k^3*x^
4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x - (4*(k^4 + 2*
k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k
^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((
a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x
^2 + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a k x + k x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*k*x + k*x**2 - 1)/(sqrt((x - 1)*(x + 1)*(k*x - 1)*(k*x + 1))*(k*x**2 + 1)), x)

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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*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {k\,x^2+a\,k\,x-1}{\left (k\,x^2+1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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