Optimal. Leaf size=208 \[ -\frac {b^3 (A (1-n)+C (2-n)) \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (3-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^2 B \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^2 C \tan (c+d x) (b \sec (c+d x))^{n-2}}{d (1-n)} \]
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Rubi [A] time = 0.22, antiderivative size = 208, normalized size of antiderivative = 1.00, 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 2643
Rule 3772
Rule 4046
Rule 4047
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*}
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Mathematica [A] time = 0.29, size = 155, normalized size = 0.75 \[ \frac {\sqrt {-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left (A (n-1) n \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\sec ^2(c+d x)\right )+(n-2) \left (B n \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\sec ^2(c+d x)\right )+C (n-1) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\sec ^2(c+d x)\right )\right )\right )}{d (n-2) (n-1) n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{2}\right )} \left (b \sec \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.52, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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