Optimal. Leaf size=124 \[ -\frac {2 \tan ^{-1}\left (\frac {(k-1) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2-2 \sqrt {k} x+1}\right )}{3 (k-1)}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {k^2+k+1} x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+\sqrt {k} x+1}\right )}{3 \sqrt {k^2+k+1}} \]
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Rubi [C] time = 3.22, antiderivative size = 610, normalized size of antiderivative = 4.92, number of steps used = 21, number of rules used = 9, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6719, 6725, 419, 2113, 537, 571, 93, 205, 208} \begin {gather*} \frac {2 \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 )}{3 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 (-1)^{2/3} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {k+\sqrt [3]{-1}} \sqrt {1-x^2}}{\sqrt {\sqrt [3]{-1} k+1} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {k+\sqrt [3]{-1}} \sqrt {\sqrt [3]{-1} k+1} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt [3]{-1} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {k-(-1)^{2/3}} \sqrt {1-x^2}}{\sqrt {1-(-1)^{2/3} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {k-(-1)^{2/3}} \sqrt {1-(-1)^{2/3} 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 {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \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 208
Rule 419
Rule 537
Rule 571
Rule 2113
Rule 6719
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1+k^{3/2} x^3}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (-1+k^{3/2} x^3\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 \left (\frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}}+\frac {2}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (-1+k^{3/2} x^3\right )}\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 (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (-1+k^{3/2} x^3\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 (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (-\frac {1}{3 \left (1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (1-(-1)^{2/3} \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 (2 \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}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \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 \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}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \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}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \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 {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\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 )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left ((-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \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 {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \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 )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+(-1)^{2/3} k-\left ((-1)^{2/3} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt [3]{-1} k-\left (-\sqrt [3]{-1} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {2 \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 )}{3 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 (-1)^{2/3} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {\sqrt [3]{-1}+k} \sqrt {1-x^2}}{\sqrt {1+\sqrt [3]{-1} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\sqrt [3]{-1}+k} \sqrt {1+\sqrt [3]{-1} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt [3]{-1} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {-(-1)^{2/3}+k} \sqrt {1-x^2}}{\sqrt {1-(-1)^{2/3} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {-(-1)^{2/3}+k} \sqrt {1-(-1)^{2/3} 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 {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 4.79, size = 444, normalized size = 3.58 \begin {gather*} \frac {2 \sqrt {k} \sqrt {x^2-1} \sqrt {k^2 x^2-1} \left (-\frac {\tanh ^{-1}\left (\frac {\sqrt {(k-1) k} \sqrt {x^2-1}}{\sqrt {1-k} \sqrt {k^2 x^2-1}}\right )}{\sqrt {1-k} \sqrt {(k-1) k}}-\frac {(-1)^{2/3} \tanh ^{-1}\left (\frac {\sqrt {k \left (k+\sqrt [3]{-1}\right )} \sqrt {x^2-1}}{\sqrt {\sqrt [3]{-1} k+1} \sqrt {k^2 x^2-1}}\right )}{\sqrt {k \left (k+\sqrt [3]{-1}\right )} \sqrt {\sqrt [3]{-1} k+1}}+\frac {\sqrt [3]{-1} \tanh ^{-1}\left (\frac {\sqrt {k \left (k-(-1)^{2/3}\right )} \sqrt {x^2-1}}{\sqrt {1-(-1)^{2/3} k} \sqrt {k^2 x^2-1}}\right )}{\sqrt {k \left (k-(-1)^{2/3}\right )} \sqrt {1-(-1)^{2/3} k}}\right )+3 \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {1}{2} i \left (i+\sqrt {3}\right ) k;\sin ^{-1}(x)|k^2\right )}{3 \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 = 4.13, size = 124, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {(-1+k) x}{1-2 \sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 (-1+k)}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {1+k+k^2} x}{1+\sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 213, normalized size = 1.72 \begin {gather*} \frac {2 \, \sqrt {k^{2} + k + 1} {\left (k - 1\right )} \arctan \left (-\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {k^{2} + k + 1} {\left ({\left (k^{2} + 2 \, k + 1\right )} x - {\left (k x^{2} + 1\right )} \sqrt {k}\right )}}{k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} + 4 \, k^{2} + 4 \, k + 1\right )} x^{2} + k}\right ) - {\left (k^{2} + k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left ({\left (k^{3} + k^{2} - k - 1\right )} x + 2 \, {\left ({\left (k^{2} - k\right )} x^{2} + k - 1\right )} \sqrt {k}\right )}}{4 \, k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} - 2 \, k^{2} + 4 \, k + 1\right )} x^{2} + 4 \, k}\right )}{3 \, {\left (k^{3} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 1538, normalized size = 12.40
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{-3+3 k}+\frac {4 \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}\, \sqrt {2}}{x \sqrt {2 k^{2}+2 k +2}}\right )}{3 \sqrt {2 k^{2}+2 k +2}}\right ) \sqrt {2}}{2}+\frac {k \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}}+\frac {k^{2} \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {-3 k^{2}}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}}-\frac {k \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}}+\frac {k^{2} \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \sqrt {-3 k^{2}}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}}+\frac {4 k^{2} \ln \left (\frac {-2 k^{2}+4 k -2+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k^{3} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )-k^{2}+2 k -1}}{x^{2}-\frac {1}{k}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {-\left (-1+k \right )^{2}}}\) | \(1538\) |
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 {2 \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticPi \left (x , k , k\right )}{3 \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {\ln \left (\frac {-2 k^{2}+4 k -2+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k^{3} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )-k^{2}+2 k -1}}{x^{2}-\frac {1}{k}}\right )}{3 \sqrt {-\left (-1+k \right )^{2}}}+\frac {\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{6 \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}}+\frac {k \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{2 \sqrt {-3 k^{2}}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{6 \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}}-\frac {k \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{2 \sqrt {-3 k^{2}}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{2} \textit {\_Z}^{4}+k \,\textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (-k \,\underline {\hspace {1.25 ex}}\alpha ^{2}-2\right ) \left (-\frac {\arctanh \left (\frac {k \left (2 k^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-k^{2}-1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} k^{4}+k^{4} x^{2}+2 k^{3} x^{2}-2 k^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+6 k^{2} x^{2}+2 k \,x^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2}-4 k^{2}+x^{2}-4 k -4\right )}{2 \left (k^{4}+2 k^{3}+6 k^{2}+2 k +1\right ) \sqrt {-k \,\underline {\hspace {1.25 ex}}\alpha ^{2} \left (k^{2}+k +1\right )}\, \sqrt {k^{3} x^{4}-k^{3} x^{2}-k \,x^{2}+k}}\right )}{\sqrt {-k \,\underline {\hspace {1.25 ex}}\alpha ^{2} \left (k^{2}+k +1\right )}}+\frac {2 k \underline {\hspace {1.25 ex}}\alpha \left (k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \sqrt {-k^{2} x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (x \sqrt {k^{2}}, -\frac {k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1}{k}, \frac {1}{\sqrt {k^{2}}}\right )}{\sqrt {k^{2}}\, \sqrt {\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right ) k}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right )}\right )}{6}}{\sqrt {k}}\) | \(1765\) |
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 {k^{\frac {3}{2}} x^{3} + 1}{{\left (k^{\frac {3}{2}} x^{3} - 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 = -0.01 \begin {gather*} \int \frac {k^{3/2}\,x^3+1}{\left (k^{3/2}\,x^3-1\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 {\left (\sqrt {k} x + 1\right ) \left (- \sqrt {k} x + k x^{2} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (\sqrt {k} x - 1\right ) \left (\sqrt {k} x + k x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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