Optimal. Leaf size=119 \[ \frac {a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {b (e \sin (c+d x))^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1)} \]
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Rubi [A] time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2838, 2564, 364, 2643} \[ \frac {a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {b (e \sin (c+d x))^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2564
Rule 2643
Rule 2838
Rule 3872
Rubi steps
\begin {align*} \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx &=-\int (-b-a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^m \, dx\\ &=a \int (e \sin (c+d x))^m \, dx+b \int \sec (c+d x) (e \sin (c+d x))^m \, dx\\ &=\frac {a \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}}{d e (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {b \operatorname {Subst}\left (\int \frac {x^m}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}\\ &=\frac {a \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}}{d e (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {b \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 98, normalized size = 0.82 \[ \frac {\tan (c+d x) (e \sin (c+d x))^m \left (a \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )+b \cos (c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.64, size = 0, normalized size = 0.00 \[ \int \left (a +b \sec \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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 \[ \int \left (e \sin {\left (c + d x \right )}\right )^{m} \left (a + b \sec {\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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