Optimal. Leaf size=50 \[ \frac {(d \tan (a+b x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2607, 364} \[ \frac {(d \tan (a+b x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2607
Rubi steps
\begin {align*} \int \cos ^2(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(d x)^n}{\left (1+x^2\right )^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\, _2F_1\left (2,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [C] time = 4.10, size = 939, normalized size = 18.78 \[ \frac {2 \left (F_1\left (\frac {n+1}{2};n,1;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 F_1\left (\frac {n+1}{2};n,2;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+4 F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \cos ^2(a+b x) \tan \left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b \left (\frac {2 (n+1) \left (-F_1\left (\frac {n+3}{2};n,2;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+8 F_1\left (\frac {n+3}{2};n,3;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-12 F_1\left (\frac {n+3}{2};n,4;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+n F_1\left (\frac {n+3}{2};n+1,1;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 n F_1\left (\frac {n+3}{2};n+1,2;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+4 n F_1\left (\frac {n+3}{2};n+1,3;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (a+b x)\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )}{n+3}+\left (F_1\left (\frac {n+1}{2};n,1;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 F_1\left (\frac {n+1}{2};n,2;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+4 F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )+n \left (F_1\left (\frac {n+1}{2};n,1;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 F_1\left (\frac {n+1}{2};n,2;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+4 F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )-2 n \left (F_1\left (\frac {n+1}{2};n,1;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 F_1\left (\frac {n+1}{2};n,2;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+4 F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec (a+b x) \tan ^2\left (\frac {1}{2} (a+b x)\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{2}, x\right ) \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.49, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (a+b\,x\right )}^2\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \]
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 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \cos ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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