Optimal. Leaf size=186 \[ -\frac {2 a b (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}}-\frac {\left (a^2 (1-n)-b^2 n\right ) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f (1-n) (1+n) \sqrt {\sin ^2(e+f x)}}+\frac {b^2 (d \cos (e+f x))^n \tan (e+f x)}{f (1-n)} \]
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Rubi [A]
time = 0.17, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4349, 3873,
3857, 2722, 4131} \begin {gather*} -\frac {\left (a^2 (1-n)-b^2 n\right ) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{f (1-n) (n+1) \sqrt {\sin ^2(e+f x)}}-\frac {2 a b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\cos ^2(e+f x)\right )}{f n \sqrt {\sin ^2(e+f x)}}+\frac {b^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rule 3873
Rule 4131
Rule 4349
Rubi steps
\begin {align*} \int (d \cos (e+f x))^n (a+b \sec (e+f x))^2 \, dx &=\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} (a+b \sec (e+f x))^2 \, dx\\ &=\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \left (a^2+b^2 \sec ^2(e+f x)\right ) \, dx+\frac {\left (2 a b (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{1-n} \, dx}{d}\\ &=\frac {b^2 (d \cos (e+f x))^n \tan (e+f x)}{f (1-n)}+\frac {\left (2 a b \left (\frac {\cos (e+f x)}{d}\right )^{-n} (d \cos (e+f x))^n\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{-1+n} \, dx}{d}+\left (\left (a^2-\frac {b^2 n}{1-n}\right ) (d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int (d \sec (e+f x))^{-n} \, dx\\ &=-\frac {2 a b (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}}+\frac {b^2 (d \cos (e+f x))^n \tan (e+f x)}{f (1-n)}+\left (\left (a^2-\frac {b^2 n}{1-n}\right ) \left (\frac {\cos (e+f x)}{d}\right )^{-n} (d \cos (e+f x))^n\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^n \, dx\\ &=-\frac {2 a b (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}}-\frac {\left (a^2-\frac {b^2 n}{1-n}\right ) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f (1+n) \sqrt {\sin ^2(e+f x)}}+\frac {b^2 (d \cos (e+f x))^n \tan (e+f x)}{f (1-n)}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 161, normalized size = 0.87 \begin {gather*} -\frac {d (d \cos (e+f x))^{-1+n} \csc (e+f x) \left (b^2 n (1+n) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\cos ^2(e+f x)\right )+a (-1+n) \cos (e+f x) \left (2 b (1+n) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(e+f x)\right )+a n \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right )\right )\right ) \sqrt {\sin ^2(e+f x)}}{f (-1+n) n (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{2}\, dx\]
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 (d \cos {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )^{2}\, 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 (d\,\cos \left (e+f\,x\right )\right )}^n\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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