Optimal. Leaf size=229 \[ -\frac{a \left (a^2 (p+2)+3 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right )}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left (a^2 \left (p^2+6 p+11\right )+2 b^2 (p+2)\right ) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2) (p+3)}-\frac{b (a+b \sin (c+d x))^2 (e \cos (c+d x))^{p+1}}{d e (p+3)}-\frac{a b (p+5) (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2) (p+3)} \]
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Rubi [A] time = 0.369708, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2692, 2862, 2669, 2643} \[ -\frac{a \left (a^2 (p+2)+3 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right )}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left (a^2 \left (p^2+6 p+11\right )+2 b^2 (p+2)\right ) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2) (p+3)}-\frac{b (a+b \sin (c+d x))^2 (e \cos (c+d x))^{p+1}}{d e (p+3)}-\frac{a b (p+5) (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2) (p+3)} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2862
Rule 2669
Rule 2643
Rubi steps
\begin{align*} \int (e \cos (c+d x))^p (a+b \sin (c+d x))^3 \, dx &=-\frac{b (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))^2}{d e (3+p)}+\frac{\int (e \cos (c+d x))^p (a+b \sin (c+d x)) \left (2 b^2+a^2 (3+p)+a b (5+p) \sin (c+d x)\right ) \, dx}{3+p}\\ &=-\frac{a b (5+p) (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))}{d e (2+p) (3+p)}-\frac{b (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))^2}{d e (3+p)}+\frac{\int (e \cos (c+d x))^p \left (a (3+p) \left (3 b^2+a^2 (2+p)\right )+b \left (2 b^2 (2+p)+a^2 \left (11+6 p+p^2\right )\right ) \sin (c+d x)\right ) \, dx}{6+5 p+p^2}\\ &=-\frac{b \left (2 b^2 (2+p)+a^2 \left (11+6 p+p^2\right )\right ) (e \cos (c+d x))^{1+p}}{d e (1+p) \left (6+5 p+p^2\right )}-\frac{a b (5+p) (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))}{d e (2+p) (3+p)}-\frac{b (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))^2}{d e (3+p)}+\left (a \left (a^2+\frac{3 b^2}{2+p}\right )\right ) \int (e \cos (c+d x))^p \, dx\\ &=-\frac{b \left (2 b^2 (2+p)+a^2 \left (11+6 p+p^2\right )\right ) (e \cos (c+d x))^{1+p}}{d e (1+p) \left (6+5 p+p^2\right )}-\frac{a \left (a^2+\frac{3 b^2}{2+p}\right ) (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)}}-\frac{a b (5+p) (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))}{d e (2+p) (3+p)}-\frac{b (e \cos (c+d x))^{1+p} (a+b \sin (c+d x))^2}{d e (3+p)}\\ \end{align*}
Mathematica [A] time = 54.7158, size = 290, normalized size = 1.27 \[ \frac{8 \sec ^2(c+d x)^{p/2} (a+b \sin (c+d x))^3 (e \cos (c+d x))^p \left (\frac{1}{3} a \left (a^2+3 b^2\right ) \tan ^3(c+d x) \, _2F_1\left (\frac{3}{2},\frac{p+4}{2};\frac{5}{2};-\tan ^2(c+d x)\right )-\frac{b \left (3 a^2+b^2\right ) \left ((p+3) \tan ^2(c+d x)+2\right ) \sec ^2(c+d x)^{-\frac{p}{2}-\frac{3}{2}}}{(p+1) (p+3)}-\frac{3 a^2 b \sec ^2(c+d x)^{-\frac{p}{2}-\frac{3}{2}}}{p+3}+a^3 \tan (c+d x) \, _2F_1\left (\frac{1}{2},\frac{p+4}{2};\frac{3}{2};-\tan ^2(c+d x)\right )\right )}{d \left (2 b \left (6 a^2+b^2\right ) \sin (2 (c+d x)) \sqrt{\sec ^2(c+d x)}+8 a^3-12 a b^2 \cos (2 (c+d x))+12 a b^2-b^3 \sin (4 (c+d x)) \sqrt{\sec ^2(c+d x)}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 3.557, 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 ) ^{3}\, 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 )}^{3} \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 (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3} \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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