Integrand size = 23, antiderivative size = 158 \[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a f}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 (a-b) f (1+p)}+\frac {(a-b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a^2 f (1+p)} \]
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Time = 0.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3751, 457, 105, 162, 67, 70} \[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {(a-b p) \left (a+b \tan ^2(e+f x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \tan ^2(e+f x)}{a}+1\right )}{2 a^2 f (p+1)}-\frac {\left (a+b \tan ^2(e+f x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \tan ^2(e+f x)+a}{a-b}\right )}{2 f (p+1) (a-b)}-\frac {\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{2 a f} \]
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Rule 67
Rule 70
Rule 105
Rule 162
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^3 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x^2 (1+x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = -\frac {\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a f}-\frac {\text {Subst}\left (\int \frac {(a+b x)^p (a-b p-b p x)}{x (1+x)} \, dx,x,\tan ^2(e+f x)\right )}{2 a f} \\ & = -\frac {\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a f}+\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}-\frac {(a-b p) \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\tan ^2(e+f x)\right )}{2 a f} \\ & = -\frac {\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a f}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 (a-b) f (1+p)}+\frac {(a-b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 a^2 f (1+p)} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\left (b+a \cot ^2(e+f x)\right ) \left (-a^2 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right )-(a-b) \left (a (1+p) \cot ^2(e+f x)+(-a+b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \tan ^2(e+f x)}{a}\right )\right )\right ) \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p}{2 a^2 (a-b) f (1+p)} \]
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\[\int \cot \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
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\[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right )^{3} \,d x } \]
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\[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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