Optimal. Leaf size=150 \[ \frac {(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {2^{n-1} (B (m-n)+i A (m+n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left (m,-n;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{f m n} \]
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Rubi [A] time = 0.23, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {3588, 79, 70, 69} \[ \frac {(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {2^{n-1} (B (m-n)+i A (m+n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left (m,-n;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{f m n} \]
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))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^{-1+m} (A+B x) (c-i c x)^{-1+n} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {(a (i B (m-n)-A (m+n))) \operatorname {Subst}\left (\int (a+i a x)^{-1+m} (c-i c x)^n \, dx,x,\tan (e+f x)\right )}{2 f n}\\ &=\frac {(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {\left (2^{-1+n} a (i B (m-n)-A (m+n)) (c-i c \tan (e+f x))^n \left (\frac {c-i c \tan (e+f x)}{c}\right )^{-n}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {i x}{2}\right )^n (a+i a x)^{-1+m} \, dx,x,\tan (e+f x)\right )}{f n}\\ &=\frac {(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {2^{-1+n} (B (m-n)+i A (m+n)) \, _2F_1\left (m,-n;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m n}\\ \end {align*}
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Mathematica [A] time = 20.93, size = 197, normalized size = 1.31 \[ \frac {2^{m+n-1} \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^n \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left ((m+1) (B-i A) \, _2F_1\left (1,-n;m+1;-e^{2 i (e+f x)}\right )-i m (A-i B) e^{2 i (e+f x)} \, _2F_1\left (1,1-n;m+2;-e^{2 i (e+f x)}\right )\right )}{f m (m+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left ({\left (A - i \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + A + i \, B\right )} \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (2 i \, f m x + 2 i \, e m + m \log \left (\frac {a}{c}\right ) + m \log \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.35, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
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 \[ \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{n} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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