Optimal. Leaf size=202 \[ \frac{(n+1)^2 (d \cot (e+f x))^{n+3} \, _2F_1\left (1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^3 f (n+3)}+\frac{i n (n+2) (d \cot (e+f x))^{n+4} \, _2F_1\left (1,\frac{n+4}{2};\frac{n+6}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^4 f (n+4)}-\frac{i n (d \cot (e+f x))^{n+3}}{4 a^2 d^3 f (\cot (e+f x)+i)}-\frac{(d \cot (e+f x))^{n+3}}{4 d^3 f (a \cot (e+f x)+i a)^2} \]
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Rubi [A] time = 0.496307, antiderivative size = 202, normalized size of antiderivative = 1., 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 = {3673, 3559, 3596, 3538, 3476, 364} \[ \frac{(n+1)^2 (d \cot (e+f x))^{n+3} \, _2F_1\left (1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^3 f (n+3)}+\frac{i n (n+2) (d \cot (e+f x))^{n+4} \, _2F_1\left (1,\frac{n+4}{2};\frac{n+6}{2};-\cot ^2(e+f x)\right )}{4 a^2 d^4 f (n+4)}-\frac{i n (d \cot (e+f x))^{n+3}}{4 a^2 d^3 f (\cot (e+f x)+i)}-\frac{(d \cot (e+f x))^{n+3}}{4 d^3 f (a \cot (e+f x)+i a)^2} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3559
Rule 3596
Rule 3538
Rule 3476
Rule 364
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 \operatorname{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)) \operatorname{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*}
Mathematica [F] time = 16.4851, size = 0, normalized size = 0. \[ \int \frac{(d \cot (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.659, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\cot \left ( fx+e \right ) \right ) ^{n}}{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} - 1}\right )^{n}{\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cot \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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