Optimal. Leaf size=72 \[ \frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{-1+n} \, _2F_1(1,-1+n;n;-i \cot (e+f x))}{f (1-n)} \]
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Rubi [A]
time = 0.13, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3754, 3624,
3618, 12, 66} \begin {gather*} \frac {a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{n-1} \, _2F_1(1,n-1;n;-i \cot (e+f x))}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 66
Rule 3618
Rule 3624
Rule 3754
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx &=d^2 \int (d \cot (e+f x))^{-2+n} (i a+a \cot (e+f x))^2 \, dx\\ &=\frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+d^2 \int (d \cot (e+f x))^{-2+n} \left (-2 a^2+2 i a^2 \cot (e+f x)\right ) \, dx\\ &=\frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+\frac {\left (4 i a^4 d^2\right ) \text {Subst}\left (\int \frac {2^{2-n} \left (-\frac {i d x}{a^2}\right )^{-2+n}}{-4 a^4-2 a^2 x} \, dx,x,2 i a^2 \cot (e+f x)\right )}{f}\\ &=\frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+\frac {\left (i 2^{4-n} a^4 d^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a^2}\right )^{-2+n}}{-4 a^4-2 a^2 x} \, dx,x,2 i a^2 \cot (e+f x)\right )}{f}\\ &=\frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{-1+n} \, _2F_1(1,-1+n;n;-i \cot (e+f x))}{f (1-n)}\\ \end {align*}
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Mathematica [F]
time = 2.64, size = 0, normalized size = 0.00 \begin {gather*} \int (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 = 0.66, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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*} - a^{2} \left (\int \left (- \left (d \cot {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \cot {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (d \cot {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx\right ) \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 {\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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