Integrand size = 23, antiderivative size = 101 \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d} \]
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Time = 0.11 (sec) , antiderivative size = 101, 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 = {3275, 525, 524} \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{3 d} \]
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Rule 524
Rule 525
Rule 3275
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (1+\frac {b x^2}{a}\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d} \\ \end{align*}
Time = 3.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d} \]
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\[\int \left (\cot ^{4}\left (d x +c \right )\right ) {\left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )}^{p}d x\]
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\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]
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\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]
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