Optimal. Leaf size=193 \[ -\frac {a \sqrt {k} \tanh ^{-1}\left (\frac {\left (2 k^{3/2}+2 \sqrt {k}\right ) x^2}{k^2 x^4+\left (k x^2-1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+2 k x^2+1}\right )}{2 (k+1)}+\frac {\tan ^{-1}\left (\frac {\left (-k-2 \sqrt {k}-1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2-1}\right )}{k+1}+\frac {\tan ^{-1}\left (\frac {\left (-k+2 \sqrt {k}-1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2-1}\right )}{k+1} \]
________________________________________________________________________________________
Rubi [C] time = 2.80, antiderivative size = 411, normalized size of antiderivative = 2.13, 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, 208} \begin {gather*} \frac {\sqrt {1-x^2} \left (2 \sqrt {-k}-a k\right ) \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} (k+1) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \left (a k+2 \sqrt {-k}\right ) \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} (k+1) \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]
[Out]
Rule 93
Rule 208
Rule 419
Rule 537
Rule 571
Rule 2113
Rule 6719
Rule 6725
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 ) \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 \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*}
________________________________________________________________________________________
Mathematica [C] time = 0.50, size = 202, normalized size = 1.05 \begin {gather*} \frac {a k \sqrt {x^2-1} \sqrt {k^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {k (k+1)} \sqrt {x^2-1}}{\sqrt {k+1} \sqrt {k^2 x^2-1}}\right )+\sqrt {k+1} \sqrt {k (k+1)} \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )-2 \sqrt {k+1} \sqrt {k (k+1)} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-k;\sin ^{-1}(x)|k^2\right )}{\sqrt {k+1} \sqrt {k (k+1)} \sqrt {\left (x^2-1\right ) \left (k^2 x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.00, size = 193, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\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 {\tan ^{-1}\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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.13, size = 1809, normalized size = 9.37
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 160, normalized size = 0.83
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________