Integrand size = 23, antiderivative size = 95 \[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a d}+\frac {(a-b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^2(c+d x)}{a}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a^2 d (1+p)} \]
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Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3273, 79, 67} \[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\frac {(a-b p) \left (a+b \sin ^2(c+d x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sin ^2(c+d x)}{a}+1\right )}{2 a^2 d (p+1)}-\frac {\csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{p+1}}{2 a d} \]
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Rule 67
Rule 79
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x) (a+b x)^p}{x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {\csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a d}-\frac {(a-b p) \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sin ^2(c+d x)\right )}{2 a d} \\ & = -\frac {\csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a d}+\frac {(a-b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^2(c+d x)}{a}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a^2 d (1+p)} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\left (a \csc ^2(c+d x)+\frac {(-a+b p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^2(c+d x)}{a}\right )}{1+p}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 a^2 d} \]
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\[\int \left (\cot ^{3}\left (d x +c \right )\right ) {\left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )}^{p}d x\]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]
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