Optimal. Leaf size=97 \[ -\frac{a \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right )}{d e (p+1) \sqrt{\sin ^2(c+d x)}}-\frac{b (e \cos (c+d x))^{p+1}}{d e (p+1)} \]
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Rubi [A] time = 0.0520769, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2669, 2643} \[ -\frac{a \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right )}{d e (p+1) \sqrt{\sin ^2(c+d x)}}-\frac{b (e \cos (c+d x))^{p+1}}{d e (p+1)} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2643
Rubi steps
\begin{align*} \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx &=-\frac{b (e \cos (c+d x))^{1+p}}{d e (1+p)}+a \int (e \cos (c+d x))^p \, dx\\ &=-\frac{b (e \cos (c+d x))^{1+p}}{d e (1+p)}-\frac{a (e \cos (c+d x))^{1+p} \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d e (1+p) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.934504, size = 240, normalized size = 2.47 \[ -\frac{(e \cos (c+d x))^p \left (-\frac{1}{2} a (p-1) \sin (2 (c+d x)) \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right )+b 2^{-p-1} \left (1+e^{2 i (c+d x)}\right ) \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^p \sqrt{\sin ^2(c+d x)} \left ((p+1) e^{i (c+d x)} \, _2F_1\left (1,\frac{p+3}{2};\frac{3-p}{2};-e^{2 i (c+d x)}\right )-(p-1) e^{-i (c+d x)} \, _2F_1\left (1,\frac{p+1}{2};\frac{1-p}{2};-e^{2 i (c+d x)}\right )\right ) \cos ^{-p}(c+d x)\right )}{\left (d-d p^2\right ) \sqrt{\sin ^2(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.875, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{p} \left ( a+b\sin \left ( dx+c \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 \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p}\,{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 \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p}, 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 (e \cos{\left (c + d x \right )}\right )^{p} \left (a + b \sin{\left (c + d 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 \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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