Integrand size = 25, antiderivative size = 70 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3308, 821, 272, 65, 214} \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 3308
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x}{x^2 \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^4(c+d x)\right )}{4 d} \\ & = -\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^4(c+d x)}\right )}{2 b d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )-\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \]
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\[\int \frac {\cot ^{3}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}d x\]
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none
Time = 0.35 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.53 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\left [\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {a} \log \left (\frac {8 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {a} + 2 \, a + b\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{4 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}}, -\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{2 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}}\right ] \]
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\[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\frac {\log \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a} - \sqrt {a}}{\sqrt {b \sin \left (d x + c\right )^{4} + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {b \sin \left (d x + c\right )^{4} + a}}{a \sin \left (d x + c\right )^{2}}}{4 \, d} \]
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Timed out. \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]
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