Optimal. Leaf size=208 \[ -\frac{b^3 (A (1-n)+C (2-n)) \sin (c+d x) (b \sec (c+d x))^{n-3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3-n}{2},\frac{5-n}{2},\cos ^2(c+d x)\right )}{d (1-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 B \sin (c+d x) (b \sec (c+d x))^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2-n}{2},\frac{4-n}{2},\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \tan (c+d x) (b \sec (c+d x))^{n-2}}{d (1-n)} \]
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Rubi [A] time = 0.216861, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {16, 4047, 3772, 2643, 4046} \[ -\frac{b^3 (A (1-n)+C (2-n)) \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 B \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \tan (c+d x) (b \sec (c+d x))^{n-2}}{d (1-n)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=b^2 \int (b \sec (c+d x))^{-2+n} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=b^2 \int (b \sec (c+d x))^{-2+n} \left (A+C \sec ^2(c+d x)\right ) \, dx+(b B) \int (b \sec (c+d x))^{-1+n} \, dx\\ &=-\frac{b^2 C (b \sec (c+d x))^{-2+n} \tan (c+d x)}{d (1-n)}+\left (b^2 \left (A+\frac{C (2-n)}{1-n}\right )\right ) \int (b \sec (c+d x))^{-2+n} \, dx+\left (b B \left (\frac{\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{1-n} \, dx\\ &=-\frac{B \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C (b \sec (c+d x))^{-2+n} \tan (c+d x)}{d (1-n)}+\left (b^2 \left (A+\frac{C (2-n)}{1-n}\right ) \left (\frac{\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{2-n} \, dx\\ &=-\frac{B \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{\left (A+\frac{C (2-n)}{1-n}\right ) \cos ^3(c+d x) \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C (b \sec (c+d x))^{-2+n} \tan (c+d x)}{d (1-n)}\\ \end{align*}
Mathematica [A] time = 0.272728, size = 155, normalized size = 0.75 \[ \frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left (A (n-1) n \cos ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-2}{2},\frac{n}{2},\sec ^2(c+d x)\right )+(n-2) \left (B n \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-1}{2},\frac{n+1}{2},\sec ^2(c+d x)\right )+C (n-1) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\sec ^2(c+d x)\right )\right )\right )}{d (n-2) (n-1) n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.056, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2}\,{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} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{2}\right )} \left (b \sec \left (d x + c\right )\right )^{n}, 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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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