Optimal. Leaf size=236 \[ \frac {a \sqrt {k} \tan ^{-1}\left (\frac {\left (2 \sqrt {k}-2 k^{3/2}\right ) x^2}{k^2 x^4+\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}-2 k x^2+1}\right )}{2 \left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )}+\frac {\tan ^{-1}\left (\frac {\left (-k-2 i \sqrt {k}+1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+1}\right )}{\left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )}+\frac {\tan ^{-1}\left (\frac {\left (-k+2 i \sqrt {k}+1\right ) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+1}\right )}{\left (\sqrt {k}-1\right ) \left (\sqrt {k}+1\right )} \]
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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.
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Rule 93
Rule 205
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+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*}
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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