Optimal. Leaf size=85 \[ -\frac {\cos ^2(a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac {1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left (\frac {1-m}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(a+b x)\right )}{b (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 85, 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 = {4309, 2576} \[ -\frac {\cos ^2(a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac {1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left (\frac {1-m}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(a+b x)\right )}{b (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2576
Rule 4309
Rubi steps
\begin {align*} \int \cos ^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 ^{2+m}(a+b x) \sin ^m(a+b x) \, dx\\ &=-\frac {\cos ^2(a+b x) \cot (a+b x) \, _2F_1\left (\frac {1-m}{2},\frac {3+m}{2};\frac {5+m}{2};\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{\frac {1-m}{2}} \sin ^m(2 a+2 b x)}{b (3+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 7.81, size = 890, normalized size = 10.47 \[ \frac {4 (m+3) \left (4 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+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 )-4 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 ) \cos ^3\left (\frac {1}{2} (a+b x)\right ) \cos ^2(a+b x) \sin \left (\frac {1}{2} (a+b x)\right ) \sin ^m(2 (a+b x))}{b (m+1) \left (8 (m+3) 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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )-2 (m+3) 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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )-8 (m+3) 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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )+2 \left (4 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+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 )-4 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+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 )-F_1\left (\frac {m+3}{2};-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 )-8 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 )-12 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 )+8 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 )+8 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 ) (\cos (a+b x)-1)\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (2 \, b x + 2 \, a\right )^{m} \cos \left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \cos \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 5.90, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (b x +a \right )\right ) \left (\sin ^{m}\left (2 b x +2 a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{m} \cos \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________