Optimal. Leaf size=270 \[ \frac{b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right )}{d^2 f (m+2) (m+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 m+7) \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2) (m+3)}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{m+1}}{d f (m+3)} \]
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Rubi [A] time = 0.379773, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2793, 3023, 2748, 2643} \[ \frac{b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right )}{d^2 f (m+2) (m+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 m+7) \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2) (m+3)}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{m+1}}{d f (m+3)} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx &=-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{1+m} (a+b \sin (e+f x))}{d f (3+m)}+\frac{\int (d \sin (e+f x))^m \left (a d \left (b^2 (1+m)+a^2 (3+m)\right )+b d \left (b^2 (2+m)+3 a^2 (3+m)\right ) \sin (e+f x)+a b^2 d (7+2 m) \sin ^2(e+f x)\right ) \, dx}{d (3+m)}\\ &=-\frac{a b^2 (7+2 m) \cos (e+f x) (d \sin (e+f x))^{1+m}}{d f (2+m) (3+m)}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{1+m} (a+b \sin (e+f x))}{d f (3+m)}+\frac{\int (d \sin (e+f x))^m \left (a d^2 (3+m) \left (3 b^2 (1+m)+a^2 (2+m)\right )+b d^2 (2+m) \left (b^2 (2+m)+3 a^2 (3+m)\right ) \sin (e+f x)\right ) \, dx}{d^2 (2+m) (3+m)}\\ &=-\frac{a b^2 (7+2 m) \cos (e+f x) (d \sin (e+f x))^{1+m}}{d f (2+m) (3+m)}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{1+m} (a+b \sin (e+f x))}{d f (3+m)}+\left (a \left (a^2+\frac{3 b^2 (1+m)}{2+m}\right )\right ) \int (d \sin (e+f x))^m \, dx+\frac{\left (b \left (b^2 (2+m)+3 a^2 (3+m)\right )\right ) \int (d \sin (e+f x))^{1+m} \, dx}{d (3+m)}\\ &=-\frac{a b^2 (7+2 m) \cos (e+f x) (d \sin (e+f x))^{1+m}}{d f (2+m) (3+m)}+\frac{a \left (a^2+\frac{3 b^2 (1+m)}{2+m}\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+m}}{d f (1+m) \sqrt{\cos ^2(e+f x)}}+\frac{b \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+m}}{d^2 f (2+m) (3+m) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{1+m} (a+b \sin (e+f x))}{d f (3+m)}\\ \end{align*}
Mathematica [A] time = 0.770993, size = 199, normalized size = 0.74 \[ \frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^m \left (\frac{b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right )}{(m+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (m+3) \left (a^2 (m+2)+3 b^2 (m+1)\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{(m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-b^2 (a+b \sin (e+f x))-\frac{a b^2 (2 m+7)}{m+2}\right )}{f (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.731, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{m} \left ( a+b\sin \left ( fx+e \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 (f x + e\right ) + a\right )}^{3} \left (d \sin \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 (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \left (d \sin \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(-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 (f x + e\right ) + a\right )}^{3} \left (d \sin \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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