Optimal. Leaf size=55 \[ -\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3230, 266, 65} \[ -\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 3230
Rubi steps
\begin {align*} \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sin ^n(c+d x)\right )}{d n}\\ &=-\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 1.00 \[ -\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, size = 0, normalized size = 0.00 \[ \int \cot \left (d x +c \right ) \left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )\,{d x} \]
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 \[ \int \mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]
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 \[ \int \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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