Optimal. Leaf size=139 \[ \frac {a \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(e+f x)\right )}{d f (m+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2748, 2643} \[ \frac {a \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(e+f x)\right )}{d f (m+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rubi steps
\begin {align*} \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx &=a \int (d \sin (e+f x))^m \, dx+\frac {b \int (d \sin (e+f x))^{1+m} \, dx}{d}\\ &=\frac {a \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 \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) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 111, normalized size = 0.80 \[ \frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) (d \sin (e+f x))^m \left (a (m+2) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(e+f x)\right )+b (m+1) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\sin ^2(e+f x)\right )\right )}{f (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.87, size = 0, normalized size = 0.00 \[ \int \left (d \sin \left (f x +e \right )\right )^{m} \left (a +b \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,\sin \left (e+f\,x\right )\right )}^m\,\left (a+b\,\sin \left (e+f\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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