Optimal. Leaf size=100 \[ \frac{C \sin (c+d x) (b \cos (c+d x))^n}{d (n+1)}-\frac{(A n+A+C n) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n (n+1) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0943137, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3014, 2643} \[ \frac{C \sin (c+d x) (b \cos (c+d x))^n}{d (n+1)}-\frac{(A n+A+C n) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n (n+1) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=b \int (b \cos (c+d x))^{-1+n} \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (b \cos (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac{(b (A+A n+C n)) \int (b \cos (c+d x))^{-1+n} \, dx}{1+n}\\ &=\frac{C (b \cos (c+d x))^n \sin (c+d x)}{d (1+n)}-\frac{(A+A n+C n) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{2+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d n (1+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.226902, size = 111, normalized size = 1.11 \[ -\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \left (A (n+2) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )+C n \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )\right )}{d n (n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.338, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sec \left ( dx+c \right ) \, 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 (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\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 (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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