Optimal. Leaf size=82 \[ \frac {\sin (a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m+2}{2};\frac {m+4}{2};\sin ^2(a+b x)\right )}{b (m+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4310, 2577} \[ \frac {\sin (a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m+2}{2};\frac {m+4}{2};\sin ^2(a+b x)\right )}{b (m+2)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 4310
Rubi steps
\begin {align*} \int \sin (a+b x) \sin ^m(2 a+2 b x) \, dx &=\left (\cos ^{-m}(a+b x) \sin ^{-m}(a+b x) \sin ^m(2 a+2 b x)\right ) \int \cos ^m(a+b x) \sin ^{1+m}(a+b x) \, dx\\ &=\frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {2+m}{2};\frac {4+m}{2};\sin ^2(a+b x)\right ) \sin (a+b x) \sin ^m(2 a+2 b x) \tan (a+b x)}{b (2+m)}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 152, normalized size = 1.85 \[ -\frac {i 2^{-m-1} e^{i (a+b x)} \left (-i e^{-2 i (a+b x)} \left (-1+e^{4 i (a+b x)}\right )\right )^{m+1} \left ((1-2 m) \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (3-2 m);e^{4 i (a+b x)}\right )+(2 m+1) e^{2 i (a+b x)} \, _2F_1\left (1,\frac {1}{4} (2 m+5);\frac {1}{4} (5-2 m);e^{4 i (a+b x)}\right )\right )}{b \left (4 m^2-1\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right ), x\right ) \]
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 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.04, size = 0, normalized size = 0.00 \[ \int \sin \left (b x +a \right ) \left (\sin ^{m}\left (2 b x +2 a \right )\right )\, dx \]
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 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )\,{d x} \]
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 \[ \int \sin \left (a+b\,x\right )\,{\sin \left (2\,a+2\,b\,x\right )}^m \,d x \]
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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