Optimal. Leaf size=196 \[ \frac {b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)}+\frac {b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 196, 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} \[ \frac {b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}}+\frac {b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)} \]
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 ^4(c+d x) \, dx &=b^4 \int (b \cos (c+d x))^{-4+n} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}-\frac {b^3 \int (b \cos (c+d x))^{-4+n} (-b (A (2-n)+C (3-n))-b B (2-n) \cos (c+d x)) \, dx}{2-n}\\ &=-\frac {b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\left (b^3 B\right ) \int (b \cos (c+d x))^{-3+n} \, dx+\frac {\left (b^4 (A (2-n)+C (3-n))\right ) \int (b \cos (c+d x))^{-4+n} \, dx}{2-n}\\ &=-\frac {b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\frac {b^3 (A (2-n)+C (3-n)) (b \cos (c+d x))^{-3+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-3+n);\frac {1}{2} (-1+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}}+\frac {b^2 B (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 (2-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 142, normalized size = 0.72 \[ -\frac {\tan (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left ((A (n-2)+C (n-3)) \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )+(n-3) \left (B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )-C \sqrt {\sin ^2(c+d x)}\right )\right )}{d (n-3) (n-2) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.40, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{4}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^4} \,d x \]
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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