Optimal. Leaf size=52 \[ -\frac {i (a+i a \tan (e+f x))^m \, _2F_1\left (2,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{4 c f m} \]
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Rubi [A] time = 0.13, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3487, 68} \[ -\frac {i (a+i a \tan (e+f x))^m \, _2F_1\left (2,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{4 c f m} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^m}{c-i c \tan (e+f x)} \, dx &=\frac {\int \cos ^2(e+f x) (a+i a \tan (e+f x))^{1+m} \, dx}{a c}\\ &=-\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{(a-x)^2} \, dx,x,i a \tan (e+f x)\right )}{c f}\\ &=-\frac {i \, _2F_1\left (2,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{4 c f m}\\ \end {align*}
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Mathematica [B] time = 16.55, size = 133, normalized size = 2.56 \[ -\frac {i 2^{m-2} \left (1+e^{2 i (e+f x)}\right )^2 \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \, _2F_1\left (1,2;m+1;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{c f m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}}{2 \, c}, x\right ) \]
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 \[ \int \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{-i \, c \tan \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.86, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{c -i c \tan \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ \frac {i \int \frac {\left (i a \tan {\left (e + f x \right )} + a\right )^{m}}{\tan {\left (e + f x \right )} + i}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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