Optimal. Leaf size=100 \[ \frac {2 \sqrt {1+2 k+k^2-c^2 k^2} \text {ArcTan}\left (\frac {\sqrt {1+2 k+k^2-c^2 k^2} x}{1+c k x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{(-1-k+c k) (1+k+c k)} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.77, antiderivative size = 1193, normalized size of antiderivative = 11.93, number of
steps used = 20, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {1976,
6860, 1117, 1738, 1230, 1720, 1261, 738, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\left (1-c^2\right ) k^2+2 k+1} x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{\sqrt {\left (1-c^2\right ) k^2+2 k+1}}+\frac {\left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \tanh ^{-1}\left (\frac {-4 \left (\left (1-c^2\right ) k^2+c \sqrt {c^2 k-4} k^{3/2}+2 k+1\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k-\sqrt {k} \sqrt {c^2 k-4}\right )^2}{4 \sqrt {2} k^{3/2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 \sqrt {2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+2}}+\frac {\left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \tanh ^{-1}\left (\frac {-2 \left (2 \left (k^2+1\right )-\left (c k+\sqrt {c^2 k-4} \sqrt {k}\right )^2\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k+\sqrt {c^2 k-4} \sqrt {k}\right )^2}{4 \sqrt {2} k^{3/2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 \sqrt {2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+2}}-\frac {\left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+4\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \Pi \left (\frac {c^2 k}{4};2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+4\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \Pi \left (\frac {c^2 k}{4};2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 212
Rule 738
Rule 1117
Rule 1230
Rule 1261
Rule 1720
Rule 1738
Rule 1976
Rule 6860
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 14.23, size = 1374, normalized size = 13.74 \begin {gather*} \frac {\frac {\sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \left (-\frac {\left (-4+\left (-4+c^2\right ) k+c^2 k^2+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{3/2} \sqrt {-4+c^2 k}\right ) \tanh ^{-1}\left (\frac {\sqrt {-2-4 k+2 \left (-1+c^2\right ) k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{5/2} \sqrt {-4+c^2 k}} \sqrt {-1+x^2}}\right )}{\sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{5/2} \sqrt {-4+c^2 k}}}+\frac {\left (-4+\left (-4+c^2\right ) k+c^2 k^2-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{3/2} \sqrt {-4+c^2 k}\right ) \tanh ^{-1}\left (\frac {\sqrt {-2-4 k+2 \left (-1+c^2\right ) k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{5/2} \sqrt {-4+c^2 k}} \sqrt {-1+x^2}}\right )}{\sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{5/2} \sqrt {-4+c^2 k}}}\right )}{\sqrt {-2+\frac {c^2 k}{2}} \sqrt {-1-2 k+\left (-1+c^2\right ) k^2}}+2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\text {ArcSin}(x)\left |k^2\right .\right )-\frac {4 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {2 c^2 k \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {8 c \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}-\frac {2 c^3 k^{3/2} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}+\frac {4 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}-\frac {2 c^2 k \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {8 c \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}-\frac {2 c^3 k^{3/2} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}}{2 \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.33, size = 11374, normalized size = 113.74
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.
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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.
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Fricas [A]
time = 0.67, 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.
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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.
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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.
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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.
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