Optimal. Leaf size=82 \[ \frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {2+m}{2};\frac {4+m}{2};\sin ^2(a+b x)\right ) \sin (a+b x) \sin ^m(2 a+2 b x) \tan (a+b x)}{b (2+m)} \]
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Rubi [A]
time = 0.05, 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 = {4395, 2657}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 4395
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] Result contains complex when optimal does not.
time = 0.57, size = 152, normalized size = 1.85 \begin {gather*} -\frac {i 2^{-1-m} e^{i (a+b x)} \left (-i e^{-2 i (a+b x)} \left (-1+e^{4 i (a+b x)}\right )\right )^{1+m} \left ((1-2 m) \, _2F_1\left (1,\frac {1}{4} (3+2 m);\frac {1}{4} (3-2 m);e^{4 i (a+b x)}\right )+e^{2 i (a+b x)} (1+2 m) \, _2F_1\left (1,\frac {1}{4} (5+2 m);\frac {1}{4} (5-2 m);e^{4 i (a+b x)}\right )\right )}{b \left (-1+4 m^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \sin \left (x b +a \right ) \left (\sin ^{m}\left (2 x b +2 a \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin {\left (a + b x \right )} \sin ^{m}{\left (2 a + 2 b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin \left (a+b\,x\right )\,{\sin \left (2\,a+2\,b\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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