Optimal. Leaf size=130 \[ -\frac {e F_1\left (1-n;\frac {1-m}{2},\frac {1}{2} (1-m-2 n);2-n;\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{\frac {1-m}{2}} \cos (c+d x) (1+\cos (c+d x))^{\frac {1}{2} (1-m-2 n)} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}}{d (1-n)} \]
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Rubi [A]
time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3961, 2965,
140, 138} \begin {gather*} -\frac {e \cos (c+d x) (1-\cos (c+d x))^{\frac {1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac {1}{2} (-m-2 n+1)} F_1\left (1-n;\frac {1-m}{2},\frac {1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 2965
Rule 3961
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n (e \sin (c+d x))^m \, dx &=\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int (-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n (e \sin (c+d x))^m \, dx\\ &=-\frac {\left (e (-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {1-m}{2}-n} (-a+a \cos (c+d x))^{\frac {1-m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \text {Subst}\left (\int (-x)^{-n} (-a-a x)^{\frac {1}{2} (-1+m)+n} (-a+a x)^{\frac {1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\left (e (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{2}-\frac {m}{2}-n} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{\frac {1-m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \text {Subst}\left (\int (-x)^{-n} (1+x)^{\frac {1}{2} (-1+m)+n} (-a+a x)^{\frac {1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\left (e (1-\cos (c+d x))^{\frac {1}{2}-\frac {m}{2}} (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{2}-\frac {m}{2}-n} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \text {Subst}\left (\int (1-x)^{\frac {1}{2} (-1+m)} (-x)^{-n} (1+x)^{\frac {1}{2} (-1+m)+n} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {e F_1\left (1-n;\frac {1-m}{2},\frac {1}{2} (1-m-2 n);2-n;\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{\frac {1-m}{2}} \cos (c+d x) (1+\cos (c+d x))^{\frac {1}{2} (1-m-2 n)} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}}{d (1-n)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(276\) vs. \(2(130)=260\).
time = 1.32, size = 276, normalized size = 2.12 \begin {gather*} \frac {4 (3+m) F_1\left (\frac {1+m}{2};n,1+m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^n \sin \left (\frac {1}{2} (c+d x)\right ) (e \sin (c+d x))^m}{d (1+m) \left ((3+m) F_1\left (\frac {1+m}{2};n,1+m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))-4 \left ((1+m) F_1\left (\frac {3+m}{2};n,2+m;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-n F_1\left (\frac {3+m}{2};1+n,1+m;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (e \sin \left (d x +c \right )\right )^{m}\, dx\]
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 (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \left (e \sin {\left (c + d x \right )}\right )^{m}\, 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 {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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