Optimal. Leaf size=132 \[ -\frac {b^4 (A (2-n)+C (3-n)) \sin (c+d x) (b \sec (c+d x))^{n-4} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right )}{d (2-n) (4-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \tan (c+d x) (b \sec (c+d x))^{n-3}}{d (2-n)} \]
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Rubi [A] time = 0.14, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {16, 4046, 3772, 2643} \[ -\frac {b^4 (A (2-n)+C (3-n)) \sin (c+d x) (b \sec (c+d x))^{n-4} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right )}{d (2-n) (4-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \tan (c+d x) (b \sec (c+d x))^{n-3}}{d (2-n)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx &=b^3 \int (b \sec (c+d x))^{-3+n} \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=-\frac {b^3 C (b \sec (c+d x))^{-3+n} \tan (c+d x)}{d (2-n)}+\left (b^3 \left (A+\frac {C (3-n)}{2-n}\right )\right ) \int (b \sec (c+d x))^{-3+n} \, dx\\ &=-\frac {b^3 C (b \sec (c+d x))^{-3+n} \tan (c+d x)}{d (2-n)}+\left (b^3 \left (A+\frac {C (3-n)}{2-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 )^{3-n} \, dx\\ &=-\frac {\left (A+\frac {C (3-n)}{2-n}\right ) \cos ^4(c+d x) \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (4-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C (b \sec (c+d x))^{-3+n} \tan (c+d x)}{d (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 118, normalized size = 0.89 \[ \frac {b \sqrt {-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{n-1} \left (A (n-1) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\sec ^2(c+d x)\right )+C (n-3) \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\sec ^2(c+d x)\right )\right )}{d (n-3) (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + A \cos \left (d x + c\right )^{3}\right )} \left (b \sec \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.90, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{3}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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