Integrand size = 25, antiderivative size = 108 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}+\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d} \]
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Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3308, 1821, 821, 272, 65, 214} \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d}+\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 3308
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^3 \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d}-\frac {\text {Subst}\left (\int \frac {4 a-(2 a-b) x}{x^2 \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{4 a d} \\ & = \frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{4 a d} \\ & = \frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^4(c+d x)\right )}{8 a d} \\ & = \frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^4(c+d x)}\right )}{4 a b d} \\ & = -\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}+\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{a d}-\frac {\csc ^4(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{4 a d} \\ \end{align*}
Time = 3.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )-4 a \csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}+b \sqrt {a+b \sin ^4(c+d x)} \left (\frac {a \csc ^4(c+d x)}{b}-\frac {\text {arctanh}\left (\sqrt {1+\frac {b \sin ^4(c+d x)}{a}}\right )}{\sqrt {1+\frac {b \sin ^4(c+d x)}{a}}}\right )}{4 a^2 d} \]
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\[\int \frac {\cot ^{5}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}d x\]
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Time = 0.39 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.44 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\left [-\frac {{\left ({\left (2 \, a - b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a - b\right )} \sqrt {a} \log \left (\frac {8 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {a} + 2 \, a + b\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} {\left (4 \, a \cos \left (d x + c\right )^{2} - 3 \, a\right )}}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}}, \frac {{\left ({\left (2 \, a - b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a - b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} {\left (4 \, a \cos \left (d x + c\right )^{2} - 3 \, a\right )}}{4 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}}\right ] \]
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\[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\frac {2 \, \sqrt {b \sin \left (d x + c\right )^{4} + a} b}{{\left (b \sin \left (d x + c\right )^{4} + a\right )} a - a^{2}} - \frac {2 \, \log \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a} - \sqrt {a}}{\sqrt {b \sin \left (d x + c\right )^{4} + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {b \log \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a} - \sqrt {a}}{\sqrt {b \sin \left (d x + c\right )^{4} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {8 \, \sqrt {b \sin \left (d x + c\right )^{4} + a}}{a \sin \left (d x + c\right )^{2}}}{8 \, d} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]
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