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 = 0.10, antiderivative size = 91, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 2643
Rule 3486
Rule 3515
Rule 3772
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*}
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Mathematica [C] time = 1.05, 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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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (b \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.69, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x +e \right )\right )^{m} \left (a +b \tan \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (b \tan \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,\cos \left (e+f\,x\right )\right )}^m\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (d \cos {\left (e + f x \right )}\right )^{m} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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