Optimal. Leaf size=125 \[ -\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}+\frac {b^2 (A (1-n)+C (2-n)) (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) (2-n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 125, 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 {b^2 (A (1-n)+C (2-n)) \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (1-n) (2-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^2 C \sin (c+d x) (b \cos (c+d x))^{n-2}}{d (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=b^3 \int (b \cos (c+d x))^{-3+n} \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}+\left (b^3 \left (A+\frac {C (2-n)}{1-n}\right )\right ) \int (b \cos (c+d x))^{-3+n} \, dx\\ &=-\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}+\frac {b^2 \left (A+\frac {C (2-n)}{1-n}\right ) (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 114, normalized size = 0.91 \begin {gather*} -\frac {(b \cos (c+d x))^n \csc (c+d x) \left (A n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right )+C (-2+n) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(c+d x)\right )\right ) \sec ^2(c+d x) \sqrt {\sin ^2(c+d x)}}{d (-2+n) n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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