Optimal. Leaf size=113 \[ -\frac {b (A+A n+C n) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{-1+n} \sin (c+d x)}{d (1-n) (1+n) \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)} \]
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Rubi [A]
time = 0.06, 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 = {4131, 3857,
2722} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rule 4131
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] Result contains complex when optimal does not.
time = 3.95, size = 262, normalized size = 2.32 \begin {gather*} -\frac {i 2^{1+n} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \left (1+e^{2 i (c+d x)}\right )^n \left (A \left (8+6 n+n^2\right ) \, _2F_1\left (\frac {n}{2},2+n;\frac {2+n}{2};-e^{2 i (c+d x)}\right )+2 (A+2 C) e^{2 i (c+d x)} n (4+n) \, _2F_1\left (\frac {2+n}{2},2+n;\frac {4+n}{2};-e^{2 i (c+d x)}\right )+A e^{4 i (c+d x)} n (2+n) \, _2F_1\left (2+n,\frac {4+n}{2};\frac {6+n}{2};-e^{2 i (c+d x)}\right )\right ) \sec ^{-2-n}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{d n (2+n) (4+n) (A+2 C+A \cos (2 c+2 d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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