Optimal. Leaf size=202 \[ -\frac {i n (d \cot (e+f x))^{3+n}}{4 a^2 d^3 f (i+\cot (e+f x))}-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}+\frac {(1+n)^2 (d \cot (e+f x))^{3+n} \, _2F_1\left (1,\frac {3+n}{2};\frac {5+n}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^3 f (3+n)}+\frac {i n (2+n) (d \cot (e+f x))^{4+n} \, _2F_1\left (1,\frac {4+n}{2};\frac {6+n}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^4 f (4+n)} \]
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Rubi [A]
time = 0.35, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3754, 3640,
3677, 3619, 3557, 371} \begin {gather*} \frac {i n (n+2) (d \cot (e+f x))^{n+4} \, _2F_1\left (1,\frac {n+4}{2};\frac {n+6}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^4 f (n+4)}+\frac {(n+1)^2 (d \cot (e+f x))^{n+3} \, _2F_1\left (1,\frac {n+3}{2};\frac {n+5}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^3 f (n+3)}-\frac {i n (d \cot (e+f x))^{n+3}}{4 a^2 d^3 f (\cot (e+f x)+i)}-\frac {(d \cot (e+f x))^{n+3}}{4 d^3 f (a \cot (e+f x)+i a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3619
Rule 3640
Rule 3677
Rule 3754
Rubi steps
\begin {align*} \int \frac {(d \cot (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx &=\frac {\int \frac {(d \cot (e+f x))^{2+n}}{(i a+a \cot (e+f x))^2} \, dx}{d^2}\\ &=-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}+\frac {\int \frac {(d \cot (e+f x))^{2+n} (-i a d (1-n)-a d (1+n) \cot (e+f x))}{i a+a \cot (e+f x)} \, dx}{4 a^2 d^3}\\ &=-\frac {i n (d \cot (e+f x))^{3+n}}{4 a^2 d^3 f (i+\cot (e+f x))}-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}+\frac {\int (d \cot (e+f x))^{2+n} \left (-2 a^2 d^2 (1+n)^2-2 i a^2 d^2 n (2+n) \cot (e+f x)\right ) \, dx}{8 a^4 d^4}\\ &=-\frac {i n (d \cot (e+f x))^{3+n}}{4 a^2 d^3 f (i+\cot (e+f x))}-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}-\frac {(1+n)^2 \int (d \cot (e+f x))^{2+n} \, dx}{4 a^2 d^2}-\frac {(i n (2+n)) \int (d \cot (e+f x))^{3+n} \, dx}{4 a^2 d^3}\\ &=-\frac {i n (d \cot (e+f x))^{3+n}}{4 a^2 d^3 f (i+\cot (e+f x))}-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}+\frac {(1+n)^2 \text {Subst}\left (\int \frac {x^{2+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{4 a^2 d f}+\frac {(i n (2+n)) \text {Subst}\left (\int \frac {x^{3+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{4 a^2 d^2 f}\\ &=-\frac {i n (d \cot (e+f x))^{3+n}}{4 a^2 d^3 f (i+\cot (e+f x))}-\frac {(d \cot (e+f x))^{3+n}}{4 d^3 f (i a+a \cot (e+f x))^2}+\frac {(1+n)^2 (d \cot (e+f x))^{3+n} \, _2F_1\left (1,\frac {3+n}{2};\frac {5+n}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^3 f (3+n)}+\frac {i n (2+n) (d \cot (e+f x))^{4+n} \, _2F_1\left (1,\frac {4+n}{2};\frac {6+n}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^4 f (4+n)}\\ \end {align*}
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Mathematica [F]
time = 16.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d \cot (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \frac {\left (d \cot \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{2}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \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*} - \frac {\int \frac {\left (d \cot {\left (e + f x \right )}\right )^{n}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \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 = -0.00 \begin {gather*} \int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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