Optimal. Leaf size=89 \[ \frac {\, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {\, _2F_1(1,m;1+m;1+i \tan (c+d x)) (a+i a \tan (c+d x))^m}{d m} \]
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Rubi [A]
time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3643, 3562, 70,
3680, 67} \begin {gather*} \frac {(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 {(a+i a \tan (c+d x))^m \, _2F_1(1,m;m+1;i \tan (c+d x)+1)}{d m} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 3562
Rule 3643
Rule 3680
Rubi steps
\begin {align*} \int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx &=i \int (a+i a \tan (c+d x))^m \, dx-\frac {i \int \cot (c+d x) (a+i a \tan (c+d x))^m (i a+a \tan (c+d x)) \, dx}{a}\\ &=\frac {a \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{x} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\, _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 {\, _2F_1(1,m;1+m;1+i \tan (c+d x)) (a+i a \tan (c+d x))^m}{d m}\\ \end {align*}
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Mathematica [F]
time = 10.67, size = 0, normalized size = 0.00 \begin {gather*} \int \cot (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 = 1.58, size = 0, normalized size = 0.00 \[\int \cot \left (d x +c \right ) \left (a +i a \tan \left (d x +c \right )\right )^{m}\, dx\]
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 {\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 )\,{\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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