Optimal. Leaf size=78 \[ \frac {d (a+i a \tan (e+f x))^m}{f m}-\frac {(i c+d) \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 f m} \]
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Rubi [A]
time = 0.05, antiderivative size = 78, 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 = {3608, 3562, 70}
\begin {gather*} \frac {d (a+i a \tan (e+f x))^m}{f m}-\frac {(d+i c) (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} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3562
Rule 3608
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x)) \, dx &=\frac {d (a+i a \tan (e+f x))^m}{f m}-(-c+i d) \int (a+i a \tan (e+f x))^m \, dx\\ &=\frac {d (a+i a \tan (e+f x))^m}{f m}-\frac {(a (i c+d)) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {d (a+i a \tan (e+f x))^m}{f m}-\frac {(i c+d) \, _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 f m}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(171\) vs. \(2(78)=156\).
time = 6.29, size = 171, normalized size = 2.19 \begin {gather*} \frac {2^{-1+m} \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left ((-i c+d) (1+m)-i (c-i d) e^{2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^m m \, _2F_1\left (1+m,1+m;2+m;-e^{2 i (e+f x)}\right )\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.64, size = 0, normalized size = 0.00 \[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )\, dx\]
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 )\, 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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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