Optimal. Leaf size=60 \[ -\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\cot ^2(c+d x)\right )}{d (n p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3659, 3476, 364} \[ -\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\cot ^2(c+d x)\right )}{d (n p+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \left (b \cot ^p(c+d x)\right )^n \, dx &=\left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \int \cot ^{n p}(c+d x) \, dx\\ &=-\frac {\left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \operatorname {Subst}\left (\int \frac {x^{n p}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\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 = 0.06, size = 58, normalized size = 0.97 \[ -\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\cot ^2(c+d x)\right )}{d n p+d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cot \left (d x + c\right )^{p}\right )^{n}, x\right ) \]
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 \[ \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (\cot ^{p}\left (d x +c \right )\right )\right )^{n}\, dx \]
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 \[ \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^p\right )}^n \,d x \]
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 \[ \int \left (b \cot ^{p}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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