Optimal. Leaf size=191 \[ \frac {b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) n \sqrt {\sin ^2(c+d x)}}-\frac {b B \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n} \]
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Rubi [A] time = 0.19, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {16, 4047, 3772, 2643, 4046} \[ \frac {b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) n \sqrt {\sin ^2(c+d x)}}-\frac {b B \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rule 4046
Rule 4047
Rubi steps
\begin {align*} \int \cos (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 \int (b \sec (c+d x))^{-1+n} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=b \int (b \sec (c+d x))^{-1+n} \left (A+C \sec ^2(c+d x)\right ) \, dx+B \int (b \sec (c+d x))^n \, dx\\ &=\frac {b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}+\frac {(b (C (-1+n)+A n)) \int (b \sec (c+d x))^{-1+n} \, dx}{n}+\left (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 )^{-n} \, dx\\ &=-\frac {B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}+\frac {\left (b (C (-1+n)+A n) \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}{n}\\ &=-\frac {B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}}+\frac {(C (1-n)-A n) \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) n \sqrt {\sin ^2(c+d x)}}+\frac {b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 161, normalized size = 0.84 \[ \frac {\sqrt {-\tan ^2(c+d x)} (b \sec (c+d x))^n \left (A n (n+1) \cos (c+d x) \cot (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\sec ^2(c+d x)\right )+(n-1) \csc (c+d x) \left (B (n+1) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\sec ^2(c+d x)\right )+C n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sec ^2(c+d x)\right )\right )\right )}{d (n-1) n (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) \sec \left (d x + c\right ) + A \cos \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.55, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (c+d\,x\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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