Optimal. Leaf size=88 \[ \frac {(1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \tan (e+f x))^{n+1} F_1(n+1;1-m,1;n+2;-i \tan (e+f x),i \tan (e+f x))}{d f (n+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3564, 135, 133} \[ \frac {(1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \tan (e+f x))^{n+1} F_1(n+1;1-m,1;n+2;-i \tan (e+f x),i \tan (e+f x))}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 3564
Rubi steps
\begin {align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {\left (i a (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n \left (1+\frac {x}{a}\right )^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {F_1(1+n;1-m,1;2+n;-i \tan (e+f x),i \tan (e+f x)) (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^{1+n} (a+i a \tan (e+f x))^m}{d f (1+n)}\\ \end {align*}
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Mathematica [F] time = 5.58, size = 0, normalized size = 0.00 \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} \left (\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.09, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \]
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 \[ \int \left (d \tan {\left (e + f x \right )}\right )^{n} \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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