Integrand size = 23, antiderivative size = 86 \[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\frac {\left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 b f (1+p)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 a f (1+p)} \]
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Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4224, 457, 81, 67} \[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)}+\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1}}{2 b f (p+1)} \]
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Rule 67
Rule 81
Rule 457
Rule 4224
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b x^2\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(-1+x) (a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = \frac {\left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 b f (1+p)}-\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = \frac {\left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 b f (1+p)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 a f (1+p)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\frac {\left (a+b \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sec ^2(e+f x)}{a}\right )\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 a b f (1+p)} \]
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\[\int \left (a +b \sec \left (f x +e \right )^{2}\right )^{p} \tan \left (f x +e \right )^{3}d x\]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{p} \tan ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \]
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