Optimal. Leaf size=72 \[ \frac{\cos (a+b x) \cos ^2(a+b x)^{n/2} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.03143, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2617} \[ \frac{\cos (a+b x) \cos ^2(a+b x)^{n/2} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \cos (a+b x) (d \tan (a+b x))^n \, dx &=\frac{\cos (a+b x) \cos ^2(a+b x)^{n/2} \, _2F_1\left (\frac{n}{2},\frac{1+n}{2};\frac{3+n}{2};\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [C] time = 2.3333, size = 452, normalized size = 6.28 \[ -\frac{2 \sin \left (\frac{1}{2} (a+b x)\right ) \cos \left (\frac{1}{2} (a+b x)\right ) \cos (a+b x) \left (F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-2 F_1\left (\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )\right ) (d \tan (a+b x))^n}{b (n+1) \left (\frac{\sec ^2\left (\frac{1}{2} (a+b x)\right ) \left ((n+3) (\cos (a+b x)+1) F_1\left (\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-(\cos (a+b x)-1) \left (F_1\left (\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-4 F_1\left (\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-n F_1\left (\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+2 n F_1\left (\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )\right )\right )}{n+3}-F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (a+b x)\right ),-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.767, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( bx+a \right ) \left ( d\tan \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan{\left (a + b x \right )}\right )^{n} \cos{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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