Optimal. Leaf size=116 \[ -\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{d}+\frac {i \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {i \, _2F_1(1,m;1+m;1+i \tan (c+d x)) (a+i a \tan (c+d x))^m}{d} \]
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Rubi [A]
time = 0.17, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3642, 3681,
3562, 70, 3680, 67} \begin {gather*} \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}-\frac {i (a+i a \tan (c+d x))^m \, _2F_1(1,m;m+1;i \tan (c+d x)+1)}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 3562
Rule 3642
Rule 3680
Rule 3681
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^m \, dx &=-\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{d}+\frac {\int \cot (c+d x) (a+i a \tan (c+d x))^m (i a m-a (1-m) \tan (c+d x)) \, dx}{a}\\ &=-\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{d}+\frac {(i m) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^m \, dx}{a}-\int (a+i a \tan (c+d x))^m \, dx\\ &=-\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{d}+\frac {(i a) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {(i a m) \text {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{x} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x) (a+i a \tan (c+d x))^m}{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}-\frac {i \, _2F_1(1,m;1+m;1+i \tan (c+d x)) (a+i a \tan (c+d x))^m}{d}\\ \end {align*}
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Mathematica [F]
time = 18.73, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.74, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{m}\, dx\]
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 \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 (c + d x \right )} - i\right )\right )^{m} \cot ^{2}{\left (c + d x \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 {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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