Optimal. Leaf size=63 \[ \frac {\, _2F_1\left (2,\frac {1}{2} (3+n p);\frac {1}{2} (5+n p);-\tan ^2(e+f x)\right ) \tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)} \]
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Rubi [A]
time = 0.08, antiderivative size = 63, 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, 2671,
371} \begin {gather*} \frac {\tan ^3(e+f x) \, _2F_1\left (2,\frac {1}{2} (n p+3);\frac {1}{2} (n p+5);-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2671
Rule 3740
Rubi steps
\begin {align*} \int \sin ^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 \sin ^2(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\left (c (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \text {Subst}\left (\int \frac {x^{2+n p}}{\left (c^2+x^2\right )^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (2,\frac {1}{2} (3+n p);\frac {1}{2} (5+n p);-\tan ^2(e+f x)\right ) \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 [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 1.67, size = 517, normalized size = 8.21 \begin {gather*} \frac {8 (6+2 n p) \left (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 )-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 ^5\left (\frac {1}{2} (e+f x)\right ) \sin ^3\left (\frac {1}{2} (e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p) \left (2 (3+n p) 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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 (3+n p) 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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (2 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 )-3 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 \left (-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 )+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 )\right ) (-1+\cos (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.64, size = 0, normalized size = 0.00 \[\int \left (\sin ^{2}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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} \sin ^{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 {\sin \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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