Optimal. Leaf size=147 \[ \frac {a B 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \, _2F_1\left (-m,-m;1-m;\frac {1}{2} (1-i \tan (e+f x))\right )}{c f m}-\frac {(B+i A) (a+i a \tan (e+f x))^{m+1} (c-i c \tan (e+f x))^{-m-1}}{2 f (m+1)} \]
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Rubi [A] time = 0.22, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {3588, 79, 70, 69} \[ \frac {a B 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \, _2F_1\left (-m,-m;1-m;\frac {1}{2} (1-i \tan (e+f x))\right )}{c f m}-\frac {(B+i A) (a+i a \tan (e+f x))^{m+1} (c-i c \tan (e+f x))^{-m-1}}{2 f (m+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 79
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^m (A+B x) (c-i c x)^{-2-m} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{2 f (1+m)}+\frac {(i a B) \operatorname {Subst}\left (\int (a+i a x)^m (c-i c x)^{-1-m} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{2 f (1+m)}+\frac {\left (i 2^m a B (a+i a \tan (e+f x))^m \left (\frac {a+i a \tan (e+f x)}{a}\right )^{-m}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^m (c-i c x)^{-1-m} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{2 f (1+m)}+\frac {2^m a B \, _2F_1\left (-m,-m;1-m;\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m}}{c f m}\\ \end {align*}
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Mathematica [A] time = 85.98, size = 177, normalized size = 1.20 \[ \frac {a e^{i (e+2 f x)} \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m (\tan (e+f x)-i) \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{-m} \sec ^{-m-1}(e+f x) (\cos (f x)+i \sin (f x))^{-m-1} (a+i a \tan (e+f x))^m \left (A+2 i B \, _2F_1\left (1,m+1;m+2;-e^{2 i (e+f x)}\right )-i B\right )}{2 c f (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left ({\left (A - i \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + A + i \, B\right )} \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{-m - 1} e^{\left (2 i \, e m + {\left (2 i \, f m + 2 i \, f\right )} x + {\left (m + 1\right )} \log \left (\frac {a}{c}\right ) + {\left (m + 1\right )} \log \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right ) + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m + 1} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 12.18, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{1+m} \left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{-1-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{m+1}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{m+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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