∫ −1+kx2 __________________________________________________(1+ckx+kx2 )∘ __________

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

Optimal. Leaf size=100 \[ \frac {2 \sqrt {-c^2 k^2+k^2+2 k+1} \tan ^{-1}\left (\frac {x \sqrt {-c^2 k^2+k^2+2 k+1}}{c k x+\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+1}\right )}{(c k-k-1) (c k+k+1)} \]

________________________________________________________________________________________

Rubi [C] time = 5.12, antiderivative size = 691, normalized size of antiderivative = 6.91, 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, 6728, 419, 2113, 537, 571, 93, 208} \begin {gather*} -\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {c^2 k-4}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (\sqrt {k} c+\sqrt {c^2 k-4}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \left (\sqrt {c^2 k-4}+c \sqrt {k}\right ) \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2} \sqrt {-c k^{3/2} \sqrt {c^2 k-4}-c^2 k^2+2 k+2}}{\sqrt {c \left (-\sqrt {k}\right ) \sqrt {c^2 k-4}+\left (2-c^2\right ) k+2} \sqrt {1-k^2 x^2}}\right )}{\sqrt {c \left (-\sqrt {k}\right ) \sqrt {c^2 k-4}+\left (2-c^2\right ) k+2} \sqrt {-c k^{3/2} \sqrt {c^2 k-4}-c^2 k^2+2 k+2} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2} \sqrt {c k^{3/2} \sqrt {c^2 k-4}-c^2 k^2+2 k+2}}{\sqrt {c \sqrt {k} \sqrt {c^2 k-4}+\left (2-c^2\right ) k+2} \sqrt {1-k^2 x^2}}\right )}{\sqrt {c \sqrt {k} \sqrt {c^2 k-4}+\left (2-c^2\right ) k+2} \sqrt {c k^{3/2} \sqrt {c^2 k-4}-c^2 k^2+2 k+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 + k*x^2)/((1 + c*k*x + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]

[Out]

((c*Sqrt[k] + Sqrt[-4 + c^2*k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTanh[(Sqrt[k]*Sqrt[2 + 2*k - c^2*k^2 - c*k^

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

)/(Sqrt[2 + (2 - c^2)*k - c*Sqrt[k]*Sqrt[-4 + c^2*k]]*Sqrt[2 + 2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k]]*Sqr

t[(1 - x^2)*(1 - k^2*x^2)]) + ((c*Sqrt[k] - Sqrt[-4 + c^2*k])*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTanh[(Sqrt[k]

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

+ c^2*k]]*Sqrt[1 - k^2*x^2])])/(Sqrt[2 + (2 - c^2)*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]]*Sqrt[2 + 2*k - c^2*k^2 + c

*k^(3/2)*Sqrt[-4 + c^2*k]]*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[

x], k^2])/Sqrt[(1 - x^2)*(1 - k^2*x^2)] - (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(4*k)/(c*Sqrt[k] - Sqrt[

-4 + c^2*k])^2, ArcSin[x], k^2])/Sqrt[(1 - x^2)*(1 - k^2*x^2)] - (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(

4*k)/(c*Sqrt[k] + Sqrt[-4 + c^2*k])^2, ArcSin[x], k^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 208

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

b}, x] && NegQ[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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-1+k x^2}{\left (1+c k x+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+k x^2}{\sqrt {1-x^2} \left (1+c k x+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+c k x}{\sqrt {1-x^2} \left (1+c k x+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+c k x}{\sqrt {1-x^2} \left (1+c k x+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 {c k-\sqrt {k} \sqrt {-4+c^2 k}}{\left (c k-\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}+\frac {c k+\sqrt {k} \sqrt {-4+c^2 k}}{\left (c k+\sqrt {k} \sqrt {-4+c^2 k}+2 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 (\sqrt {k} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (c k-\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \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 {k} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (c k+\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \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 {\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 k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 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 (k \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 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 (2 k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 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 (k \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 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 {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{\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 {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 k^2-\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-\left (4 k^2-k^2 \left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 k^2-\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-\left (4 k^2-k^2 \left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {2+2 k-c^2 k^2-c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {1-x^2}}{\sqrt {2+\left (2-c^2\right ) k-c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {1-k^2 x^2}}\right )}{\sqrt {2+\left (2-c^2\right ) k-c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {2+2 k-c^2 k^2-c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {2+2 k-c^2 k^2+c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {1-x^2}}{\sqrt {2+\left (2-c^2\right ) k+c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {1-k^2 x^2}}\right )}{\sqrt {2+\left (2-c^2\right ) k+c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {2+2 k-c^2 k^2+c k^{3/2} \sqrt {-4+c^2 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 {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 8.33, size = 2156, normalized size = 21.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((k^(3/2)*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*(((c*Sqrt[k] - Sqrt[-4 + c^2*k])*ArcTanh[(Sqrt[k^2*(2 + 2*k - c^2

*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])]*Sqrt[-1 + x^2])/(Sqrt[k*(2 - (-2 + c^2)*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])]*S

qrt[-1 + k^2*x^2])])/(Sqrt[k*(2 - (-2 + c^2)*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])]*Sqrt[k^2*(2 + 2*k - c^2*k^2 + c*

k^(3/2)*Sqrt[-4 + c^2*k])]) + ((c*Sqrt[k] + Sqrt[-4 + c^2*k])*ArcTanh[(Sqrt[-(k^2*(-2 - 2*k + c^2*k^2 + c*k^(3

/2)*Sqrt[-4 + c^2*k]))]*Sqrt[-1 + x^2])/(Sqrt[-(k*(-2 + (-2 + c^2)*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]))]*Sqrt[-1 +

k^2*x^2])])/(Sqrt[-(k*(-2 + (-2 + c^2)*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]))]*Sqrt[-(k^2*(-2 - 2*k + c^2*k^2 + c*k

^(3/2)*Sqrt[-4 + c^2*k]))])))/Sqrt[(-1 + x^2)*(-1 + k^2*x^2)]) + (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[Ar

cSin[x], k^2])/Sqrt[(-1 + x^2)*(-1 + k^2*x^2)] - (4*c^2*k*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/

(2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])*(

-1/2*(-2*k + c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/k^2 + (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2)

)*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)]) + (c^4*k^2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*k - c^2*k

^2 - c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])*(-1/2*(-2*k +

c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/k^2 + (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2))*Sqrt[(-1 +

x^2)*(-1 + k^2*x^2)]) - (2*c*Sqrt[k]*Sqrt[-4 + c^2*k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*

k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])*(-1/

2*(-2*k + c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/k^2 + (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2))*S

qrt[(-1 + x^2)*(-1 + k^2*x^2)]) + (c^3*k^(3/2)*Sqrt[-4 + c^2*k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2

*k^2)/(2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2

*k])*(-1/2*(-2*k + c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/k^2 + (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(

2*k^2))*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)]) - (4*c^2*k*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*k -

c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])*((-2*k

+ c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2) - (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2))*Sqrt[

(-1 + x^2)*(-1 + k^2*x^2)]) + (c^4*k^2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*k - c^2*k^2 + c*

k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])*((-2*k + c^2*k^2 - c

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

1 + k^2*x^2)]) + (2*c*Sqrt[k]*Sqrt[-4 + c^2*k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*k - c^2*

k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])*((-2*k + c^2

*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2) - (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2))*Sqrt[(-1 +

x^2)*(-1 + k^2*x^2)]) - (c^3*k^(3/2)*Sqrt[-4 + c^2*k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k^2)/(2*

k - c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/((2*k - c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])*((-2

*k + c^2*k^2 - c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2) - (-2*k + c^2*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])/(2*k^2))*Sq

rt[(-1 + x^2)*(-1 + k^2*x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 3.72, size = 100, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+2 k+k^2-c^2 k^2} \tan ^{-1}\left (\frac {\sqrt {1+2 k+k^2-c^2 k^2} x}{1+c k x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{(-1-k+c k) (1+k+c k)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[1 + 2*k + k^2 - c^2*k^2]*ArcTan[(Sqrt[1 + 2*k + k^2 - c^2*k^2]*x)/(1 + c*k*x + k*x^2 + Sqrt[1 + (-1 -

k^2)*x^2 + k^2*x^4])])/((-1 - k + c*k)*(1 + k + c*k))

________________________________________________________________________________________

fricas [A] time = 0.94, size = 342, normalized size = 3.42 \begin {gather*} \left [\frac {\log \left (-\frac {{\left ({\left (2 \, c^{2} - 1\right )} k^{4} - 2 \, k^{3} - k^{2}\right )} x^{4} + 2 \, {\left (c k^{4} + 2 \, c k^{3} + c k^{2}\right )} x^{3} + {\left (2 \, c^{2} - 1\right )} k^{2} - {\left ({\left (c^{2} - 2\right )} k^{4} - 2 \, {\left (c^{2} + 3\right )} k^{3} + {\left (c^{2} - 8\right )} k^{2} - 6 \, k - 2\right )} x^{2} + 2 \, \sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left (c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x\right )} \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1} + 2 \, {\left (c k^{3} + 2 \, c k^{2} + c k\right )} x - 2 \, k - 1}{2 \, c k^{2} x^{3} + k^{2} x^{4} + 2 \, c k x + {\left (c^{2} k^{2} + 2 \, k\right )} x^{2} + 1}\right )}{2 \, \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}}, -\frac {\sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1}}{c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x}\right )}{{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}\right ] \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log(-(((2*c^2 - 1)*k^4 - 2*k^3 - k^2)*x^4 + 2*(c*k^4 + 2*c*k^3 + c*k^2)*x^3 + (2*c^2 - 1)*k^2 - ((c^2 - 2

)*k^4 - 2*(c^2 + 3)*k^3 + (c^2 - 8)*k^2 - 6*k - 2)*x^2 + 2*sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(c*k^2*x^2 + c*k

+ (k^2 + 2*k + 1)*x)*sqrt((c^2 - 1)*k^2 - 2*k - 1) + 2*(c*k^3 + 2*c*k^2 + c*k)*x - 2*k - 1)/(2*c*k^2*x^3 + k^2

*x^4 + 2*c*k*x + (c^2*k^2 + 2*k)*x^2 + 1))/sqrt((c^2 - 1)*k^2 - 2*k - 1), -sqrt(-(c^2 - 1)*k^2 + 2*k + 1)*arct

an(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*sqrt(-(c^2 - 1)*k^2 + 2*k + 1)/(c*k^2*x^2 + c*k + (k^2 + 2*k + 1)*x))/((c

^2 - 1)*k^2 - 2*k - 1)]

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k x^{2} - 1}{{\left (c k x + 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((k*x^2-1)/(c*k*x+k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B] time = 0.40, size = 4734, normalized size = 47.34


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(4734\)
default	\(\text {Expression too large to display}\)	\(11374\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^3*k^2/(c^2*k^3*(c^2*k-4))^(1/2)*2^(1/2)/((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c

^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/

k^2)^(1/2)*ln(((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*

k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2-(c^2*k^3*(c^2

*k-4))^(1/2)-2*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^2*k^4-(c^2*k^

3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k

+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)

^2+4*(-k^2-1+c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+2*(

k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2

*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4)

)^(1/2)+2*k)/k^2))+1/4*c*2^(1/2)/((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^

3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*ln(

((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c

^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)-2

*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(

1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^

3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^2+4*(-k^2-1+c

^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+2*(k^4*c^4-c^2*k^

4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4)

)^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k

^2))+c*k/(c^2*k^3*(c^2*k-4))^(1/2)*2^(1/2)/((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k

^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)

^(1/2)*ln(((k^4*c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+

2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2-(c^2*k^3*(c^2*k-4

))^(1/2)-2*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^2*k^4-(c^2*k^3*(c

^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k

^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^2+4

*(-k^2-1+c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)+2*(k^4*

c^4-c^2*k^4-(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3+2*(c^2*k^3

*(c^2*k-4))^(1/2)*k+4*k^2+(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2+1/2*(-c^2*k^2+(c^2*k^3*(c^2*k-4))^(1

/2)+2*k)/k^2))+1/4*c^3*k^2/(c^2*k^3*(c^2*k-4))^(1/2)*2^(1/2)/((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k

^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4

))^(1/2)+2*k)/k^2)^(1/2)*ln(((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^

2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2

+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^

2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*

k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/

2)-2*k)/k^2)^2+4*(-k^2-1+c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*

k)/k^2)+2*(k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2

*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2-1/2*(c^2*k^2+(c^2*k^3

*(c^2*k-4))^(1/2)-2*k)/k^2))+1/4*c*2^(1/2)/((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k

^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)

^(1/2)*ln(((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+

2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2+(c^2*k^3*(c^2*k-4

))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^2*k^4+(c^2*k^3*(c^

2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^

2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)^2+4*(

-k^2-1+c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)+2*(k^4*c^4

-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c

^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-

2*k)/k^2))-c*k/(c^2*k^3*(c^2*k-4))^(1/2)*2^(1/2)/((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3

-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k

)/k^2)^(1/2)*ln(((k^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2

)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2+(-k^2-1+c^2*k^2+(c^2*k^3*(c

^2*k-4))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)+1/2*2^(1/2)*((k^4*c^4-c^2*k^4+(c^2*k

^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*k^3*(c^2*k-4))^(1/2)*

k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2)*(4*k^2*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)

^2+4*(-k^2-1+c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)*(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^(1/2)-2*k)/k^2)+2*(k

^4*c^4-c^2*k^4+(c^2*k^3*(c^2*k-4))^(1/2)*c^2*k^2-4*c^2*k^3-c^2*k^2-(c^2*k^3*(c^2*k-4))^(1/2)*k^2+2*k^3-2*(c^2*

k^3*(c^2*k-4))^(1/2)*k+4*k^2-(c^2*k^3*(c^2*k-4))^(1/2)+2*k)/k^2)^(1/2))/(x^2-1/2*(c^2*k^2+(c^2*k^3*(c^2*k-4))^

(1/2)-2*k)/k^2))-1/(2*c^2*k^2-2*k^2-4*k-2)^(1/2)*arctanh(((-x^2+1)*(-k^2*x^2+1))^(1/2)*2^(1/2)/x/(2*c^2*k^2-2*

k^2-4*k-2)^(1/2))*2^(1/2)

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*k-4>0)', see `assume?` for

more details)Is c^2*k-4 positive, negative or zero?

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]