Optimal. Leaf size=62 \[ -\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\cot ^2(c+d x)\right )}{d (1+n p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557,
371} \begin {gather*} -\frac {\cot (c+d x) \, _2F_1\left (1,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\cot ^2(c+d x)\right ) \left (a (b \cot (c+d x))^p\right )^n}{d (n p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \left (a (b \cot (c+d x))^p\right )^n \, dx &=\left ((b \cot (c+d x))^{-n p} \left (a (b \cot (c+d x))^p\right )^n\right ) \int (b \cot (c+d x))^{n p} \, dx\\ &=-\frac {\left (b (b \cot (c+d x))^{-n p} \left (a (b \cot (c+d x))^p\right )^n\right ) \text {Subst}\left (\int \frac {x^{n p}}{b^2+x^2} \, dx,x,b \cot (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\cot ^2(c+d x)\right )}{d (1+n p)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.97 \begin {gather*} -\frac {\cot (c+d x) \left (a (b \cot (c+d x))^p\right )^n \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\cot ^2(c+d x)\right )}{d+d n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \left (a \left (b \cot \left (d x +c \right )\right )^{p}\right )^{n}\, dx\]
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 (a \left (b \cot {\left (c + d x \right )}\right )^{p}\right )^{n}\, 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 = -0.02 \begin {gather*} \int {\left (a\,{\left (b\,\mathrm {cot}\left (c+d\,x\right )\right )}^p\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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