Integrand size = 35, antiderivative size = 861 \[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=-\frac {\sqrt {a-b+c} \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}+\frac {\sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{c} (a-b+c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) (a-b+c) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
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Time = 0.67 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3781, 1325, 1295, 1211, 1117, 1209, 1230, 1720} \[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=-\frac {\sqrt {a-b+c} \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}\right )}{2 e}-\frac {\cot (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e}+\frac {\sqrt {c} \tan (d+e x) \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}{e \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac {\sqrt [4]{c} (a-b+c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{2 \sqrt [4]{a} e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) (a-b+c) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right ) \sqrt {\frac {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}{\left (\sqrt {c} \tan ^2(d+e x)+\sqrt {a}\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {c \tan ^4(d+e x)+b \tan ^2(d+e x)+a}} \]
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Rule 1117
Rule 1209
Rule 1211
Rule 1230
Rule 1295
Rule 1325
Rule 1720
Rule 3781
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2+c x^4}}{x^2 \left (1+x^2\right )} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {a+c x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}-\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}-\frac {\text {Subst}\left (\int \frac {-a c-a c x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{a e}-\frac {\left (\sqrt {a} (a-b+c)\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) e}+\frac {\left (\sqrt {c} (a-b+c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) e} \\ & = -\frac {\sqrt {a-b+c} \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}+\frac {\sqrt [4]{c} (a-b+c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) (a-b+c) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a} \sqrt {c}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\left (\left (\sqrt {a}+\sqrt {c}\right ) \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {\sqrt {a-b+c} \arctan \left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\cot (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e}+\frac {\sqrt {c} \tan (d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{e \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \sqrt [4]{c} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\sqrt [4]{c} (a-b+c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (\sqrt {a}+\sqrt {c}\right ) (a-b+c) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}},2 \arctan \left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.25 (sec) , antiderivative size = 1258, normalized size of antiderivative = 1.46 \[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {\sqrt {\frac {3 a+b+3 c+4 a \cos (2 (d+e x))-4 c \cos (2 (d+e x))+a \cos (4 (d+e x))-b \cos (4 (d+e x))+c \cos (4 (d+e x))}{3+4 \cos (2 (d+e x))+\cos (4 (d+e x))}} \left (-\cot (d+e x)+\frac {1}{2} \sin (2 (d+e x))\right )}{e}+\frac {i \sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right ) \left (1+\tan ^2(d+e x)\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}-2 i \sqrt {2} c \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \left (1+\tan ^2(d+e x)\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}+2 i \sqrt {2} a \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \left (1+\tan ^2(d+e x)\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}-2 i \sqrt {2} b \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \left (1+\tan ^2(d+e x)\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}+2 i \sqrt {2} c \operatorname {EllipticPi}\left (\frac {b+\sqrt {b^2-4 a c}}{2 c},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \left (1+\tan ^2(d+e x)\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c \tan ^2(d+e x)}{b-\sqrt {b^2-4 a c}}}-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x) \left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )}{4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \left (1+\tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
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\[\int \cot \left (e x +d \right )^{2} \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}d x\]
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Timed out. \[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\text {Timed out} \]
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\[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \cot ^{2}{\left (d + e x \right )}\, dx \]
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\[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{2} \,d x } \]
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\[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{2} \,d x } \]
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Timed out. \[ \int \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^2\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]
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