Optimal. Leaf size=84 \[ \frac {\sin ^2(a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m+3}{2};\frac {m+5}{2};\sin ^2(a+b x)\right )}{b (m+3)} \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4310, 2577} \[ \frac {\sin ^2(a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m+3}{2};\frac {m+5}{2};\sin ^2(a+b x)\right )}{b (m+3)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 4310
Rubi steps
\begin {align*} \int \sin ^2(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 ^{2+m}(a+b x) \, dx\\ &=\frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {3+m}{2};\frac {5+m}{2};\sin ^2(a+b x)\right ) \sin ^2(a+b x) \sin ^m(2 a+2 b x) \tan (a+b x)}{b (3+m)}\\ \end {align*}
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Mathematica [C] time = 3.51, size = 602, normalized size = 7.17 \[ \frac {16 (m+3) \sin ^3\left (\frac {1}{2} (a+b x)\right ) \cos ^5\left (\frac {1}{2} (a+b x)\right ) \sin ^m(2 (a+b x)) \left (F_1\left (\frac {m+1}{2};-m,2 (m+1);\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-F_1\left (\frac {m+1}{2};-m,2 m+3;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )}{b (m+1) \left (-2 (m+3) \cos ^2\left (\frac {1}{2} (a+b x)\right ) F_1\left (\frac {m+1}{2};-m,2 m+3;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 (\cos (a+b x)-1) \left (m F_1\left (\frac {m+3}{2};1-m,2 (m+1);\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-m F_1\left (\frac {m+3}{2};1-m,2 m+3;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-2 m F_1\left (\frac {m+3}{2};-m,2 (m+2);\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-3 F_1\left (\frac {m+3}{2};-m,2 (m+2);\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 m F_1\left (\frac {m+3}{2};-m,2 m+3;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 F_1\left (\frac {m+3}{2};-m,2 m+3;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )+(m+3) (\cos (a+b x)+1) F_1\left (\frac {m+1}{2};-m,2 (m+1);\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{m}, x\right ) \]
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 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.23, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{2}\left (b x +a \right )\right ) \left (\sin ^{m}\left (2 b x +2 a \right )\right )\, dx \]
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 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )^{2}\,{d x} \]
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 \[ \int {\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^m \,d x \]
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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