Optimal. Leaf size=191 \[ \frac {b^2 (C (1-n)-A n) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{-2+n} \sin (c+d x)}{d (2-n) n \sqrt {\sin ^2(c+d x)}}-\frac {b B \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{-1+n} \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n} \]
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Rubi [A]
time = 0.13, 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, 4132, 3857,
2722, 4131} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3857
Rule 4131
Rule 4132
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 = 0.40, size = 161, normalized size = 0.84 \begin {gather*} \frac {\left (A n (1+n) \cos (c+d x) \cot (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\sec ^2(c+d x)\right )+(-1+n) \csc (c+d x) \left (B (1+n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\sec ^2(c+d x)\right )+C n \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sec ^2(c+d x)\right )\right )\right ) (b \sec (c+d x))^n \sqrt {-\tan ^2(c+d x)}}{d (-1+n) n (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.39, 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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \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 \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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