Optimal. Leaf size=96 \[ -\frac {b (d \cot (e+f x))^n \, _2F_1\left (1,\frac {n}{2};\frac {2+n}{2};-\cot ^2(e+f x)\right )}{f n}-\frac {a (d \cot (e+f x))^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)} \]
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Rubi [A]
time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3754, 3619,
3557, 371} \begin {gather*} -\frac {a (d \cot (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)}-\frac {b (d \cot (e+f x))^n \, _2F_1\left (1,\frac {n}{2};\frac {n+2}{2};-\cot ^2(e+f x)\right )}{f n} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3619
Rule 3754
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n (a+b \tan (e+f x)) \, dx &=d \int (d \cot (e+f x))^{-1+n} (b+a \cot (e+f x)) \, dx\\ &=a \int (d \cot (e+f x))^n \, dx+(b d) \int (d \cot (e+f x))^{-1+n} \, dx\\ &=-\frac {(a d) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {x^{-1+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=-\frac {b (d \cot (e+f x))^n \, _2F_1\left (1,\frac {n}{2};\frac {2+n}{2};-\cot ^2(e+f x)\right )}{f n}-\frac {a (d \cot (e+f x))^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 88, normalized size = 0.92 \begin {gather*} -\frac {(d \cot (e+f x))^n \left (b (1+n) \, _2F_1\left (1,\frac {n}{2};\frac {2+n}{2};-\cot ^2(e+f x)\right )+a n \cot (e+f x) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )\right )}{f n (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +b \tan \left (f x +e \right )\right )\, dx\]
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 (d \cot {\left (e + f x \right )}\right )^{n} \left (a + b \tan {\left (e + f x \right )}\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 {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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