3.27.48 \(\int \frac {1+a k x+k x^2}{(-1+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\)

Optimal. Leaf size=236 \[ \frac {a \sqrt {k} \tan ^{-1}\left (\frac {\left (2 \sqrt {k}-2 k^{3/2}\right ) x^2}{k^2 x^4+\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}-2 k x^2+1}\right )}{2 \left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )}+\frac {\tan ^{-1}\left (\frac {\left (-k-2 i \sqrt {k}+1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+1}\right )}{\left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )}+\frac {\tan ^{-1}\left (\frac {\left (-k+2 i \sqrt {k}+1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+1}\right )}{\left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )} \]

________________________________________________________________________________________

Rubi [C]  time = 2.38, antiderivative size = 389, normalized size of antiderivative = 1.65, number of steps used = 16, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {6719, 6725, 419, 2113, 537, 571, 93, 205} \begin {gather*} -\frac {\sqrt {1-x^2} \left (2-a \sqrt {k}\right ) \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \left (a \sqrt {k}+2\right ) \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \left (2-a \sqrt {k}\right ) \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 {\sqrt {1-x^2} \left (a \sqrt {k}+2\right ) \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 {\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 )}} \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]

-1/2*((2 - a*Sqrt[k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTan[(Sqrt[k]*Sqrt[1 - x^2])/Sqrt[1 - k^2*x^2]])/((1 -
k)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + ((2 + a*Sqrt[k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTan[(Sqrt[k]*Sqrt[1 -
x^2])/Sqrt[1 - k^2*x^2]])/(2*(1 - k)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Ellipt
icF[ArcSin[x], k^2])/Sqrt[(1 - x^2)*(1 - k^2*x^2)] - ((2 - a*Sqrt[k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Elliptic
Pi[k, ArcSin[x], k^2])/(2*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) - ((2 + a*Sqrt[k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*El
lipticPi[k, ArcSin[x], k^2])/(2*Sqrt[(1 - x^2)*(1 - k^2*x^2)])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 2113

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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+a \sqrt {k}}{2 \left (1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {2-a \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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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-a \sqrt {k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 (1-k) \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} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{2 (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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.46, size = 204, normalized size = 0.86 \begin {gather*} \frac {a k \sqrt {x^2-1} \sqrt {k^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {-((k-1) k)} \sqrt {x^2-1}}{\sqrt {k-1} \sqrt {k^2 x^2-1}}\right )+\sqrt {k-1} \sqrt {-((k-1) k)} \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )-2 \sqrt {k-1} \sqrt {-((k-1) k)} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{\sqrt {k-1} \sqrt {-((k-1) k)} \sqrt {\left (x^2-1\right ) \left (k^2 x^2-1\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*k*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*ArcTanh[(Sqrt[-((-1 + k)*k)]*Sqrt[-1 + x^2])/(Sqrt[-1 + k]*Sqrt[-1 + k^
2*x^2])] + Sqrt[-1 + k]*Sqrt[-((-1 + k)*k)]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - 2*Sqrt
[-1 + k]*Sqrt[-((-1 + k)*k)]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[k, ArcSin[x], k^2])/(Sqrt[-1 + k]*Sqrt
[-((-1 + k)*k)]*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.64, size = 236, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\left (1-2 i \sqrt {k}-k\right ) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )}+\frac {\tan ^{-1}\left (\frac {\left (1+2 i \sqrt {k}-k\right ) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )}+\frac {a \sqrt {k} \tan ^{-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 \left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[((1 - (2*I)*Sqrt[k] - k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/((-1 + Sqrt[k])*(1 + Sqrt
[k])) + ArcTan[((1 + (2*I)*Sqrt[k] - k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/((-1 + Sqrt[k])*(
1 + Sqrt[k])) + (a*Sqrt[k]*ArcTan[((2*Sqrt[k] - 2*k^(3/2))*x^2)/(1 - 2*k*x^2 + k^2*x^4 + (1 + k*x^2)*Sqrt[1 +
(-1 - k^2)*x^2 + k^2*x^4])])/(2*(-1 + Sqrt[k])*(1 + Sqrt[k]))

________________________________________________________________________________________

fricas [B]  time = 2.04, size = 1793, normalized size = 7.60

result too large to display

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)*sqr
t(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))

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [A]  time = 0.20, size = 171, normalized size = 0.72

method	result	size
elliptic	\(-\frac {a \ln \left (\frac {-\frac {2 \left (k^{2}-2 k +1\right )}{k}+\left (-k^{2}+2 k -1\right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\frac {k^{2}-2 k +1}{k}}\, \sqrt {k^{2} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{2}+2 k -1\right ) \left (x^{2}-\frac {1}{k}\right )-\frac {k^{2}-2 k +1}{k}}}{x^{2}-\frac {1}{k}}\right )}{2 \sqrt {-\frac {k^{2}-2 k +1}{k}}}+\frac {\arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{-1+k}\)	\(171\)
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {a \arctanh \left (-\frac {k \,x^{2}}{\sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {k}{2 \sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {k^{2} x^{2}}{2 \sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {1}{2 \sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}\, k}+\frac {x^{2}}{2 \sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {1}{\sqrt {-\frac {1}{k}+2-k}\, \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )}{2 \sqrt {-\frac {1}{k}+2-k}}-\frac {2 \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticPi \left (x , k , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\)	\(364\)

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]

-1/2*a/(-(k^2-2*k+1)/k)^(1/2)*ln((-2*(k^2-2*k+1)/k+(-k^2+2*k-1)*(x^2-1/k)+2*(-(k^2-2*k+1)/k)^(1/2)*(k^2*(x^2-1
/k)^2+(-k^2+2*k-1)*(x^2-1/k)-(k^2-2*k+1)/k)^(1/2))/(x^2-1/k))+1/(-1+k)*arctan(((-x^2+1)*(-k^2*x^2+1))^(1/2)/x/
(-1+k))

________________________________________________________________________________________

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \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)

________________________________________________________________________________________

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)

________________________________________________________________________________________