Optimal. Leaf size=200 \[ -\frac{\left (a^2 (2-m)+b^2 (1-m)\right ) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right )}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-1}(c+d x)}{d (1-m)}-\frac{2 a b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right )}{d (2-m) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.188567, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3238, 3788, 3772, 2643, 4046} \[ -\frac{\left (a^2 (2-m)+b^2 (1-m)\right ) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right )}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-1}(c+d x)}{d (1-m)}-\frac{2 a b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right )}{d (2-m) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3788
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \sec ^m(c+d x) \, dx &=\int \sec ^{-2+m}(c+d x) (b+a \sec (c+d x))^2 \, dx\\ &=(2 a b) \int \sec ^{-1+m}(c+d x) \, dx+\int \sec ^{-2+m}(c+d x) \left (b^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=-\frac{a^2 \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (1-m)}+\left (b^2+\frac{a^2 (2-m)}{1-m}\right ) \int \sec ^{-2+m}(c+d x) \, dx+\left (2 a b \cos ^m(c+d x) \sec ^m(c+d x)\right ) \int \cos ^{1-m}(c+d x) \, dx\\ &=-\frac{a^2 \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (1-m)}-\frac{2 a b \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right ) \sec ^{-2+m}(c+d x) \sin (c+d x)}{d (2-m) \sqrt{\sin ^2(c+d x)}}+\left (\left (b^2+\frac{a^2 (2-m)}{1-m}\right ) \cos ^m(c+d x) \sec ^m(c+d x)\right ) \int \cos ^{2-m}(c+d x) \, dx\\ &=-\frac{a^2 \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (1-m)}-\frac{\left (b^2+\frac{a^2 (2-m)}{1-m}\right ) \, _2F_1\left (\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right ) \sec ^{-3+m}(c+d x) \sin (c+d x)}{d (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right ) \sec ^{-2+m}(c+d x) \sin (c+d x)}{d (2-m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.308575, size = 159, normalized size = 0.8 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-3}(c+d x) \left (a (m-2) \sec ^2(c+d x) \left (a (m-1) \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right )+2 b m \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(c+d x)\right )\right )+b^2 (m-1) m \, _2F_1\left (\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(c+d x)\right )\right )}{d (m-2) (m-1) m} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.473, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( dx+c \right ) \right ) ^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sec \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cos{\left (c + d x \right )}\right )^{2} \sec ^{m}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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