Optimal. Leaf size=119 \[ \frac {a \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {b \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m)} \]
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Rubi [A]
time = 0.11, 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 = {3957, 2917,
2644, 371, 2722} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2644
Rule 2722
Rule 2917
Rule 3957
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 \text {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 = 0.07, size = 98, normalized size = 0.82 \begin {gather*} \frac {\left (a \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right )+b \cos (c+d x) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right )\right ) (e \sin (c+d x))^m \tan (c+d x)}{d (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, 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 \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 \left (e \sin {\left (c + d x \right )}\right )^{m} \left (a + b \sec {\left (c + d x \right )}\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 {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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