Optimal. Leaf size=69 \[ -\frac {(b \cos (c+d x))^{2+n} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {16, 2722}
\begin {gather*} -\frac {\sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rubi steps
\begin {align*} \int \cos (c+d x) (b \cos (c+d x))^n \, dx &=\frac {\int (b \cos (c+d x))^{1+n} \, dx}{b}\\ &=-\frac {(b \cos (c+d x))^{2+n} \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 70, normalized size = 1.01 \begin {gather*} -\frac {\cos (c+d x) (b \cos (c+d x))^n \cot (c+d x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{n}\, dx\]
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 (b \cos {\left (c + d x \right )}\right )^{n} \cos {\left (c + d 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 (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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