Optimal. Leaf size=113 \[ \frac {C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac {b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4046, 3772, 2643} \[ \frac {C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac {b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}+\frac {(A+A n+C n) \int (b \sec (c+d x))^n \, dx}{1+n}\\ &=\frac {C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}+\frac {\left ((A+A n+C 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 )^{-n} \, dx}{1+n}\\ &=-\frac {(A+A n+C n) \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 \left (1-n^2\right ) \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}\\ \end {align*}
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Mathematica [C] time = 6.26, size = 273, normalized size = 2.42 \[ -\frac {i 2^{n+1} e^{-i (n+1) (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+1} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (n e^{i (n+2) (c+d x)} \left (2 (n+4) (A+2 C) \, _2F_1\left (1,-\frac {n}{2};\frac {n+4}{2};-e^{2 i (c+d x)}\right )+A (n+2) e^{2 i (c+d x)} \, _2F_1\left (1,1-\frac {n}{2};\frac {n+6}{2};-e^{2 i (c+d x)}\right )\right )+A \left (n^2+6 n+8\right ) e^{i n (c+d x)} \, _2F_1\left (1,-\frac {n}{2}-1;\frac {n+2}{2};-e^{2 i (c+d x)}\right )\right )}{d n (n+2) (n+4) (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.53, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}\,{d x} \]
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 \[ \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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