Optimal. Leaf size=117 \[ \frac {C (b \cos (c+d x))^{3+n} \sin (c+d x)}{b^3 d (4+n)}-\frac {(C (3+n)+A (4+n)) (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) (4+n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {16, 3093, 2722}
\begin {gather*} \frac {C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b^3 d (n+4)}-\frac {(A (n+4)+C (n+3)) \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {n+5}{2};\cos ^2(c+d x)\right )}{b^3 d (n+3) (n+4) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {\int (b \cos (c+d x))^{2+n} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac {C (b \cos (c+d x))^{3+n} \sin (c+d x)}{b^3 d (4+n)}+\frac {\left (A+\frac {C (3+n)}{4+n}\right ) \int (b \cos (c+d x))^{2+n} \, dx}{b^2}\\ &=\frac {C (b \cos (c+d x))^{3+n} \sin (c+d x)}{b^3 d (4+n)}-\frac {\left (A+\frac {C (3+n)}{4+n}\right ) (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 122, normalized size = 1.04 \begin {gather*} -\frac {\cos ^2(c+d x) (b \cos (c+d x))^n \cot (c+d x) \left (A (5+n) \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right )+C (3+n) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {5+n}{2};\frac {7+n}{2};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (3+n) (5+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\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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