Integrand size = 23, antiderivative size = 180 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (15 a^2-b (10 a-b (3-2 p)) (1-2 p)\right ) \cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 a^2 f} \]
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Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3744, 473, 464, 372, 371} \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\left (15-\frac {b (1-2 p) (10 a-b (3-2 p))}{a^2}\right ) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right )}{15 f}-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{5 a f} \]
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Rule 371
Rule 372
Rule 464
Rule 473
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2 \left (a+b x^2\right )^p}{x^6} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}+\frac {\text {Subst}\left (\int \frac {\left (10 a-b (3-2 p)+5 a x^2\right ) \left (a+b x^2\right )^p}{x^4} \, dx,x,\tan (e+f x)\right )}{5 a f} \\ & = -\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (-15 a^2-b (10 a-b (3-2 p)) (-3+2 (1+p))\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 a^2 f} \\ & = -\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (\left (-15 a^2-b (10 a-b (3-2 p)) (-3+2 (1+p))\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 a^2 f} \\ & = -\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (15 a^2-b (10 a-b (3-2 p)) (1-2 p)\right ) \cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 a^2 f} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cot (e+f x) \left (3 \cot ^4(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-p,-\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a}\right )+10 \cot ^2(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-p,-\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right )+15 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right )\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 f} \]
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\[\int \csc \left (f x +e \right )^{6} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
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\[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6} \,d x } \]
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Timed out. \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6} \,d x } \]
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\[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6} \,d x } \]
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Timed out. \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^6} \,d x \]
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