Optimal. Leaf size=65 \[ \frac {\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)}+\frac {\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3659, 2607, 14} \[ \frac {\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)}+\frac {\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 3659
Rubi steps
\begin {align*} \int \sec ^4(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \sec ^4(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname {Subst}\left (\int (c x)^{n p} \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname {Subst}\left (\int \left ((c x)^{n p}+\frac {(c x)^{2+n p}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}+\frac {\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)}\\ \end {align*}
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Mathematica [A] time = 2.13, size = 87, normalized size = 1.34 \[ \frac {\cot (e+f x) \left (2 \left (-\tan ^2(e+f x)\right )^{\frac {1}{2} (1-n p)}+\tan ^2(e+f x) \left ((n p+1) \sec ^2(e+f x)+2\right )\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1) (n p+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 75, normalized size = 1.15 \[ \frac {{\left (n p + 2 \, \cos \left (f x + e\right )^{2} + 1\right )} e^{\left (n p \log \left (\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \relax (b)\right )} \sin \left (f x + e\right )}{{\left (f n^{2} p^{2} + 4 \, f n p + 3 \, f\right )} \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.14, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{4}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 71, normalized size = 1.09 \[ \frac {\frac {b^{p} c^{n p} {\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )^{3}}{n p + 3} + \frac {b^{p} c^{n p} {\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )}{n p + 1}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\cos \left (e+f\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \sec ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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