Optimal. Leaf size=78 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f \sqrt {a-b}}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \]
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Rubi [A] time = 0.12, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3670, 480, 12, 377, 203} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f \sqrt {a-b}}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 377
Rule 480
Rule 3670
Rubi steps
\begin {align*} \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f}-\frac {\operatorname {Subst}\left (\int \frac {a}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f}-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {a-b} f}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f}\\ \end {align*}
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Mathematica [C] time = 9.50, size = 212, normalized size = 2.72 \[ -\frac {\cos ^2(e+f x) \cot (e+f x) \left (\frac {b \tan ^2(e+f x)}{a}+1\right ) \left (\frac {4 \sin ^2(e+f x) \left (a^2+a b \left (\tan ^2(e+f x)-1\right )-b^2 \tan ^2(e+f x)\right ) \, _2F_1\left (2,2;\frac {5}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )}{3 a^2}+\frac {\sin ^{-1}\left (\sqrt {\frac {(a-b) \sin ^2(e+f x)}{a}}\right ) \left (a+2 b \tan ^2(e+f x)\right )}{a \sqrt {\frac {\sin ^2(e+f x) \cos ^2(e+f x) \left (a^2+a b \left (\tan ^2(e+f x)-1\right )-b^2 \tan ^2(e+f x)\right )}{a^2}}}\right )}{f \sqrt {a+b \tan ^2(e+f x)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 289, normalized size = 3.71 \[ \left [-\frac {a \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) + 4 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a - b\right )}}{4 \, {\left (a^{2} - a b\right )} f \tan \left (f x + e\right )}, -\frac {\sqrt {a - b} a \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right ) + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a - b\right )}}{2 \, {\left (a^{2} - a b\right )} f \tan \left (f x + e\right )}\right ] \]
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 \[ \int \frac {\cot \left (f x + e\right )^{2}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.35, size = 1195, normalized size = 15.32 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot \left (f x + e\right )^{2}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}}\,{d x} \]
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 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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