Optimal. Leaf size=114 \[ -\frac {i F_1\left (m;-n,1;1+m;-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}}{2 f m} \]
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Rubi [A]
time = 0.13, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3645, 142, 141}
\begin {gather*} -\frac {i (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n} F_1\left (m;-n,1;m+1;-\frac {d (i \tan (e+f x)+1)}{i c-d},\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 3645
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \left (c-\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {\left (i a^2 (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \left (\frac {c}{c+i d}-\frac {i d x}{a (c+i d)}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=-\frac {i F_1\left (m;-n,1;1+m;-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}}{2 f m}\\ \end {align*}
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Mathematica [F]
time = 7.82, size = 0, normalized size = 0.00 \begin {gather*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.58, size = 0, normalized size = 0.00 \[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{n}\, dx\]
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 \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 (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{n}\, 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 (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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