Optimal. Leaf size=72 \[ \frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} \, _2F_1\left (\frac {n}{2},\frac {1+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {2697}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 2.54, size = 452, normalized size = 6.28 \begin {gather*} -\frac {2 \left (F_1\left (\frac {1+n}{2};n,1;\frac {3+n}{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 {1+n}{2};n,2;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \cos \left (\frac {1}{2} (a+b x)\right ) \cos (a+b x) \sin \left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b (1+n) \left (-F_1\left (\frac {1+n}{2};n,1;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+\frac {\left (-\left (\left (F_1\left (\frac {3+n}{2};n,2;\frac {5+n}{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 {3+n}{2};n,3;\frac {5+n}{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 {3+n}{2};1+n,1;\frac {5+n}{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 {3+n}{2};1+n,2;\frac {5+n}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) (-1+\cos (a+b x))\right )+(3+n) F_1\left (\frac {1+n}{2};n,2;\frac {3+n}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (1+\cos (a+b x))\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )}{3+n}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \cos \left (b x +a \right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx\]
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 \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 (d \tan {\left (a + b x \right )}\right )^{n} \cos {\left (a + b 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 = -0.01 \begin {gather*} \int \cos \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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