Optimal. Leaf size=105 \[ -\frac {i 2^{n-\frac {m}{2}} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} \, _2F_1\left (-\frac {m}{2},\frac {1}{2} (m-2 n+2);1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]
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Rubi [A] time = 0.23, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3515, 3505, 3523, 70, 69} \[ -\frac {i 2^{n-\frac {m}{2}} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} \, _2F_1\left (-\frac {m}{2},\frac {1}{2} (m-2 n+2);1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3515
Rule 3523
Rubi steps
\begin {align*} \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx &=\left ((e \cos (c+d x))^m (e \sec (c+d x))^m\right ) \int (e \sec (c+d x))^{-m} (a+i a \tan (c+d x))^n \, dx\\ &=\left ((e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \int (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-\frac {m}{2}+n} \, dx\\ &=\frac {\left (a^2 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \operatorname {Subst}\left (\int (a-i a x)^{-1-\frac {m}{2}} (a+i a x)^{-1-\frac {m}{2}+n} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{-1-\frac {m}{2}+n} a (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^n \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {m}{2}-n}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-1-\frac {m}{2}+n} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {i 2^{-\frac {m}{2}+n} (e \cos (c+d x))^m \, _2F_1\left (-\frac {m}{2},\frac {1}{2} (2+m-2 n);1-\frac {m}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{2} (m-2 n)} (a+i a \tan (c+d x))^n}{d m}\\ \end {align*}
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Mathematica [A] time = 13.32, size = 192, normalized size = 1.83 \[ \frac {i 2^{n-m} \left (1+e^{2 i (c+d x)}\right ) \left (e^{i d x}\right )^n \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \cos ^{-m}(c+d x) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m \, _2F_1\left (1,\frac {m+2}{2};-\frac {m}{2}+n+1;-e^{2 i (c+d x)}\right )}{d (m-2 n)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} e^{\left (i \, d m x + i \, c m + m \log \left (a e\right ) - m \log \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{m} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.06, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{m} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (e \cos {\left (c + d x \right )}\right )^{m} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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