Optimal. Leaf size=122 \[ \frac {\, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 (i c+d) f m}-\frac {d \, _2F_1\left (1,m;1+m;-\frac {d (1+i \tan (e+f x))}{i c-d}\right ) (a+i a \tan (e+f x))^m}{\left (c^2+d^2\right ) f m} \]
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Rubi [A]
time = 0.18, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3643, 3562, 70,
3680} \begin {gather*} \frac {(a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m (d+i c)}-\frac {d (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;-\frac {d (i \tan (e+f x)+1)}{i c-d}\right )}{f m \left (c^2+d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3562
Rule 3643
Rule 3680
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx &=\frac {\int (a+i a \tan (e+f x))^m \, dx}{c-i d}-\frac {d \int \frac {(a+i a \tan (e+f x))^m (i a+a \tan (e+f x))}{c+d \tan (e+f x)} \, dx}{a (c-i d)}\\ &=\frac {a \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{(i c+d) f}+\frac {(a d) \text {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{c+d x} \, dx,x,\tan (e+f x)\right )}{(i c+d) f}\\ &=\frac {\, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 (i c+d) f m}-\frac {d \, _2F_1\left (1,m;1+m;-\frac {d (1+i \tan (e+f x))}{i c-d}\right ) (a+i a \tan (e+f x))^m}{\left (c^2+d^2\right ) f m}\\ \end {align*}
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Mathematica [F]
time = 14.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.23, size = 0, normalized size = 0.00 \[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{c +d \tan \left (f x +e \right )}\, dx\]
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 \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{c + d \tan {\left (e + f x \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 \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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