Optimal. Leaf size=194 \[ \frac {b^2 (A (1-n)+C (2-n)) \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (1-n) (2-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^2 C \sin (c+d x) (b \cos (c+d x))^{n-2}}{d (1-n)}+\frac {b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 3023, 2748, 2643} \[ \frac {b^2 (A (1-n)+C (2-n)) \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (1-n) (2-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^2 C \sin (c+d x) (b \cos (c+d x))^{n-2}}{d (1-n)}+\frac {b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=b^3 \int (b \cos (c+d x))^{-3+n} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}-\frac {b^2 \int (b \cos (c+d x))^{-3+n} (-b (A (1-n)+C (2-n))-b B (1-n) \cos (c+d x)) \, dx}{1-n}\\ &=-\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}+\left (b^2 B\right ) \int (b \cos (c+d x))^{-2+n} \, dx+\frac {\left (b^3 (A (1-n)+C (2-n))\right ) \int (b \cos (c+d x))^{-3+n} \, dx}{1-n}\\ &=-\frac {b^2 C (b \cos (c+d x))^{-2+n} \sin (c+d x)}{d (1-n)}+\frac {b^2 (A (1-n)+C (2-n)) (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {b B (b \cos (c+d x))^{-1+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 137, normalized size = 0.71 \[ -\frac {b \tan (c+d x) (b \cos (c+d x))^{n-1} \left ((A (n-1)+C (n-2)) \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )+(n-2) \left (B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\cos ^2(c+d x)\right )-C \sqrt {\sin ^2(c+d x)}\right )\right )}{d (n-2) (n-1) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.62, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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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