Optimal. Leaf size=127 \[ \frac {b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)} \]
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Rubi [A] time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {16, 3014, 2643} \[ \frac {b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}}-\frac {b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int (b \cos (c+d x))^n \left (A+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+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)}+\left (b^4 \left (A+\frac {C (3-n)}{2-n}\right )\right ) \int (b \cos (c+d x))^{-4+n} \, dx\\ &=-\frac {b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\frac {b^3 \left (A+\frac {C (3-n)}{2-n}\right ) (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 (3-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 122, normalized size = 0.96 \[ -\frac {\sqrt {\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^n \left (A (n-1) \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )+C (n-3) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\cos ^2(c+d x)\right )\right )}{d (n-3) (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{4}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^4} \,d x \]
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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