Optimal. Leaf size=90 \[ -\frac{a \sin (e+f x) (d \cos (e+f x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{d f (m+1) \sqrt{\sin ^2(e+f x)}}-\frac{b (d \cos (e+f x))^m}{f m} \]
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Rubi [A] time = 0.0999522, antiderivative size = 91, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3515, 3486, 3772, 2643} \[ -\frac{a \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^m \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{f (m+1) \sqrt{\sin ^2(e+f x)}}-\frac{b (d \cos (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (d \cos (e+f x))^m (a+b \tan (e+f x)) \, dx &=\left ((d \cos (e+f x))^m (d \sec (e+f x))^m\right ) \int (d \sec (e+f x))^{-m} (a+b \tan (e+f x)) \, dx\\ &=-\frac{b (d \cos (e+f x))^m}{f m}+\left (a (d \cos (e+f x))^m (d \sec (e+f x))^m\right ) \int (d \sec (e+f x))^{-m} \, dx\\ &=-\frac{b (d \cos (e+f x))^m}{f m}+\left (a \left (\frac{\cos (e+f x)}{d}\right )^{-m} (d \cos (e+f x))^m\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^m \, dx\\ &=-\frac{b (d \cos (e+f x))^m}{f m}-\frac{a \cos (e+f x) (d \cos (e+f x))^m \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f (1+m) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.02678, size = 203, normalized size = 2.26 \[ \frac{(d \cos (e+f x))^m \left (-a (m-2) m \sin (2 (e+f x)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )-2 b \left (m^2-m-2\right ) \sqrt{\sin ^2(e+f x)} \, _2F_1\left (1,\frac{m}{2};1-\frac{m}{2};-e^{2 i (e+f x)}\right )+2 b m (m+1) \sqrt{\sin ^2(e+f x)} \, _2F_1\left (1,\frac{m+2}{2};2-\frac{m}{2};-e^{2 i (e+f x)}\right ) (\cos (2 (e+f x))+i \sin (2 (e+f x)))\right )}{2 f (m-2) m (m+1) \sqrt{\sin ^2(e+f x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.558, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{m} \left ( a+b\tan \left ( fx+e \right ) \right ) \, 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 \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\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 \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\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 (d \cos{\left (e + f x \right )}\right )^{m} \left (a + b \tan{\left (e + f x \right )}\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 \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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