Optimal. Leaf size=49 \[ -\frac {i (a+i a \tan (c+d x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d m} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3481, 68} \[ -\frac {i (a+i a \tan (c+d x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d m} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3481
Rubi steps
\begin {align*} \int (a+i a \tan (c+d x))^m \, dx &=-\frac {(i a) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}\\ \end {align*}
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Mathematica [B] time = 0.48, size = 128, normalized size = 2.61 \[ -\frac {i 2^{m-1} \left (1+e^{2 i (c+d x)}\right ) \left (e^{i d x}\right )^m \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \, _2F_1\left (1,1;m+1;-e^{2 i (c+d x)}\right ) \sec ^{-m}(c+d x) (\cos (d x)+i \sin (d x))^{-m} (a+i a \tan (c+d x))^m}{d m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.20, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \]
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 \[ \int \left (i a \tan {\left (c + d x \right )} + a\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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