Integrand size = 21, antiderivative size = 129 \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {a \cos ^2(c+d x)^{\frac {2+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {2+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)} \]
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Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3969, 3557, 371, 2697} \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\frac {a (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d e (m+1)}+\frac {a \sec (c+d x) \cos ^2(c+d x)^{\frac {m+2}{2}} (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {m+2}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)} \]
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Rule 371
Rule 2697
Rule 3557
Rule 3969
Rubi steps \begin{align*} \text {integral}& = a \int (e \tan (c+d x))^m \, dx+a \int \sec (c+d x) (e \tan (c+d x))^m \, dx \\ & = \frac {a \cos ^2(c+d x)^{\frac {2+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {2+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {(a e) \text {Subst}\left (\int \frac {x^m}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{d} \\ & = \frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {a \cos ^2(c+d x)^{\frac {2+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {2+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\frac {a (e \tan (c+d x))^m \left (\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)}{1+m}+\csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3}{2},\sec ^2(c+d x)\right ) \left (-\tan ^2(c+d x)\right )^{\frac {1-m}{2}}\right )}{d} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right ) \left (e \tan \left (d x +c \right )\right )^{m}d x\]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=a \left (\int \left (e \tan {\left (c + d x \right )}\right )^{m}\, dx + \int \left (e \tan {\left (c + d x \right )}\right )^{m} \sec {\left (c + d x \right )}\, dx\right ) \]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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