Integrand size = 23, antiderivative size = 136 \[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)}-\frac {\csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-p,-\frac {2-n}{n},-\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^p \left (1+\frac {b \sin ^n(c+d x)}{a}\right )^{-p}}{2 d} \]
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Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3309, 1858, 372, 371, 272, 67} \[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)}-\frac {\csc ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \left (\frac {b \sin ^n(c+d x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-p,-\frac {2-n}{n},-\frac {b \sin ^n(c+d x)}{a}\right )}{2 d} \]
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Rule 67
Rule 272
Rule 371
Rule 372
Rule 1858
Rule 3309
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (a+b x^n\right )^p}{x^3} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (a+b x^n\right )^p}{x^3}-\frac {\left (a+b x^n\right )^p}{x}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x^3} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\text {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sin ^n(c+d x)\right )}{d n}+\frac {\left (\left (a+b \sin ^n(c+d x)\right )^p \left (1+\frac {b \sin ^n(c+d x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^n}{a}\right )^p}{x^3} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\operatorname {Hypergeometric2F1}\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)}-\frac {\csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-p,-\frac {2-n}{n},-\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^p \left (1+\frac {b \sin ^n(c+d x)}{a}\right )^{-p}}{2 d} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\frac {\left (a+b \sin ^n(c+d x)\right )^p \left (\frac {2 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )}{a n (1+p)}-\csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-p,\frac {-2+n}{n},-\frac {b \sin ^n(c+d x)}{a}\right ) \left (1+\frac {b \sin ^n(c+d x)}{a}\right )^{-p}\right )}{2 d} \]
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\[\int \left (\cot ^{3}\left (d x +c \right )\right ) {\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p}d x\]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]
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