Optimal. Leaf size=232 \[ \frac {\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{a d e (m+1) \sqrt {\cos ^2(c+d x)}}-\frac {b e (e \sin (c+d x))^{m-1} \left (-\frac {a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right )^{\frac {1-m}{2}} \left (\frac {a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right )^{\frac {1-m}{2}} F_1\left (1-m;\frac {1-m}{2},\frac {1-m}{2};2-m;-\frac {a-b}{b+a \cos (c+d x)},\frac {a+b}{b+a \cos (c+d x)}\right )}{a^2 d (1-m)} \]
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Rubi [A] time = 0.26, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3872, 2867, 2643, 2703} \[ \frac {\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{a d e (m+1) \sqrt {\cos ^2(c+d x)}}-\frac {b e (e \sin (c+d x))^{m-1} \left (-\frac {a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right )^{\frac {1-m}{2}} \left (\frac {a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right )^{\frac {1-m}{2}} F_1\left (1-m;\frac {1-m}{2},\frac {1-m}{2};2-m;-\frac {a-b}{b+a \cos (c+d x)},\frac {a+b}{b+a \cos (c+d x)}\right )}{a^2 d (1-m)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2703
Rule 2867
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^m}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) (e \sin (c+d x))^m}{-b-a \cos (c+d x)} \, dx\\ &=\frac {\int (e \sin (c+d x))^m \, dx}{a}+\frac {b \int \frac {(e \sin (c+d x))^m}{-b-a \cos (c+d x)} \, dx}{a}\\ &=-\frac {b e F_1\left (1-m;\frac {1-m}{2},\frac {1-m}{2};2-m;-\frac {a-b}{b+a \cos (c+d x)},\frac {a+b}{b+a \cos (c+d x)}\right ) \left (-\frac {a (1-\cos (c+d x))}{b+a \cos (c+d x)}\right )^{\frac {1-m}{2}} \left (\frac {a (1+\cos (c+d x))}{b+a \cos (c+d x)}\right )^{\frac {1-m}{2}} (e \sin (c+d x))^{-1+m}}{a^2 d (1-m)}+\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{a d e (1+m) \sqrt {\cos ^2(c+d x)}}\\ \end {align*}
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Mathematica [B] time = 5.97, size = 687, normalized size = 2.96 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) (e \sin (c+d x))^m \left ((a+b) \, _2F_1\left (\frac {m+1}{2},m+1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b F_1\left (\frac {m+1}{2};m,1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )\right )}{d (a+b \sec (c+d x)) \left (2 m \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left ((a+b) \, _2F_1\left (\frac {m+1}{2},m+1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b F_1\left (\frac {m+1}{2};m,1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )\right )+2 m \tan \left (\frac {1}{2} (c+d x)\right ) \cot (c+d x) \left ((a+b) \, _2F_1\left (\frac {m+1}{2},m+1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b F_1\left (\frac {m+1}{2};m,1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left ((a+b) \, _2F_1\left (\frac {m+1}{2},m+1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b F_1\left (\frac {m+1}{2};m,1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )\right )+\frac {(m+1) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {2 b \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left ((b-a) F_1\left (\frac {m+3}{2};m,2;\frac {m+5}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )+m (a+b) F_1\left (\frac {m+3}{2};m+1,1;\frac {m+5}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}\right )\right )}{m+3}-(a+b)^2 \left (\, _2F_1\left (\frac {m+1}{2},m+1;\frac {m+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )^{-m-1}\right )\right )}{a+b}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \sin \left (d x + c\right )\right )^{m}}{b \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{b \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x +c \right )\right )^{m}}{a +b \sec \left (d x +c \right )}\, dx \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{b \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^m}{b+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e \sin {\left (c + d x \right )}\right )^{m}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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