Integrand size = 23, antiderivative size = 243 \[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\frac {3 a^3 (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {a^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {3 a^3 \cos ^2(c+d x)^{\frac {2+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {2+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {a^3 \cos ^2(c+d x)^{\frac {4+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {4+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec ^3(c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)} \]
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Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3971, 3557, 371, 2697, 2687, 32} \[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\frac {a^3 (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d e (m+1)}+\frac {a^3 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac {m+4}{2}} (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {m+4}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}+\frac {3 a^3 \sec (c+d x) \cos ^2(c+d x)^{\frac {m+2}{2}} (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {m+2}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}+\frac {3 a^3 (e \tan (c+d x))^{m+1}}{d e (m+1)} \]
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Rule 32
Rule 371
Rule 2687
Rule 2697
Rule 3557
Rule 3971
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 (e \tan (c+d x))^m+3 a^3 \sec (c+d x) (e \tan (c+d x))^m+3 a^3 \sec ^2(c+d x) (e \tan (c+d x))^m+a^3 \sec ^3(c+d x) (e \tan (c+d x))^m\right ) \, dx \\ & = a^3 \int (e \tan (c+d x))^m \, dx+a^3 \int \sec ^3(c+d x) (e \tan (c+d x))^m \, dx+\left (3 a^3\right ) \int \sec (c+d x) (e \tan (c+d x))^m \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) (e \tan (c+d x))^m \, dx \\ & = \frac {3 a^3 \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^3 \cos ^2(c+d x)^{\frac {4+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {4+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec ^3(c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int (e x)^m \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (a^3 e\right ) \text {Subst}\left (\int \frac {x^m}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{d} \\ & = \frac {3 a^3 (e \tan (c+d x))^{1+m}}{d e (1+m)}+\frac {a^3 \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 {3 a^3 \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^3 \cos ^2(c+d x)^{\frac {4+m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {4+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec ^3(c+d x) (e \tan (c+d x))^{1+m}}{d e (1+m)} \\ \end{align*}
Time = 2.10 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.93 \[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\frac {a^3 e (e \tan (c+d x))^{-1+m} \left (-\tan ^2(c+d x)\right )^{-m/2} \left (9 \sqrt {-\tan ^2(c+d x)}+9 (1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3}{2},\sec ^2(c+d x)\right ) \sec (c+d x) \sqrt {-\tan ^2(c+d x)}+(1+m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-m}{2},\frac {5}{2},\sec ^2(c+d x)\right ) \sec ^3(c+d x) \sqrt {-\tan ^2(c+d x)}-9 \left (-\tan ^2(c+d x)\right )^{\frac {2+m}{2}}-3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \left (-\tan ^2(c+d x)\right )^{\frac {2+m}{2}}\right )}{3 d (1+m)} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right )^{3} \left (e \tan \left (d x +c \right )\right )^{m}d x\]
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\[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=a^{3} \left (\int \left (e \tan {\left (c + d x \right )}\right )^{m}\, dx + \int 3 \left (e \tan {\left (c + d x \right )}\right )^{m} \sec {\left (c + d x \right )}\, dx + \int 3 \left (e \tan {\left (c + d x \right )}\right )^{m} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (e \tan {\left (c + d x \right )}\right )^{m} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3 \,d x \]
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