Optimal. Leaf size=61 \[ \frac {\, _2F_1\left (2,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)} \]
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Rubi [A]
time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3740, 2687,
371} \begin {gather*} \frac {\tan (e+f x) \, _2F_1\left (2,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2687
Rule 3740
Rubi steps
\begin {align*} \int \cos ^2(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 \cos ^2(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 ) \text {Subst}\left (\int \frac {(c x)^{n p}}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (2,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 3.42, size = 1060, normalized size = 17.38 \begin {gather*} \frac {2 \left (F_1\left (\frac {1}{2} (1+n p);n p,1;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {1}{2} (1+n p);n p,2;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+4 F_1\left (\frac {1}{2} (1+n p);n p,3;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^2(e+f x) \tan \left (\frac {1}{2} (e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f \left (\left (F_1\left (\frac {1}{2} (1+n p);n p,1;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {1}{2} (1+n p);n p,2;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+4 F_1\left (\frac {1}{2} (1+n p);n p,3;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )+n p \left (F_1\left (\frac {1}{2} (1+n p);n p,1;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {1}{2} (1+n p);n p,2;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+4 F_1\left (\frac {1}{2} (1+n p);n p,3;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)+\frac {2 (1+n p) \left (-F_1\left (\frac {1}{2} (3+n p);n p,2;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+8 F_1\left (\frac {1}{2} (3+n p);n p,3;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-12 F_1\left (\frac {1}{2} (3+n p);n p,4;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n p F_1\left (\frac {1}{2} (3+n p);1+n p,1;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 n p F_1\left (\frac {1}{2} (3+n p);1+n p,2;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+4 n p F_1\left (\frac {1}{2} (3+n p);1+n p,3;\frac {1}{2} (5+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{3+n p}-2 n p \left (F_1\left (\frac {1}{2} (1+n p);n p,1;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {1}{2} (1+n p);n p,2;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+4 F_1\left (\frac {1}{2} (1+n p);n p,3;\frac {1}{2} (3+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec (e+f x) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cos ^{2}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\cos \left (e+f\,x\right )}^2\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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