Integrand size = 23, antiderivative size = 71 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \]
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Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3274, 331, 211} \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \]
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Rule 211
Rule 331
Rule 3274
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\cot ^3(c+d x)}{3 a d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d} \\ & = \frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {3 (a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \cot (c+d x) \left (4 a+3 b-a \csc ^2(c+d x)\right )}{3 a^{5/2} d} \]
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Time = 2.65 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {-a -b}{a^{2} \tan \left (d x +c \right )}}{d}\) | \(79\) |
default | \(\frac {\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {-a -b}{a^{2} \tan \left (d x +c \right )}}{d}\) | \(79\) |
risch | \(\frac {2 i \left (6 a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a +3 b \right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 a^{2} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b}{2 a^{3} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 a^{2} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b}{2 a^{3} d}\) | \(275\) |
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 402, normalized size of antiderivative = 5.66 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {4 \, {\left (4 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) - 12 \, {\left (a + b\right )} \cos \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {2 \, {\left (4 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 6 \, {\left (a + b\right )} \cos \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} + \frac {3 \, {\left (a + b\right )} \tan \left (d x + c\right )^{2} - a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\sqrt {a^{2} + a b} a^{2}} + \frac {3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} - a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 13.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,{\left (a+b\right )}^{3/2}}{a^{5/2}\,d}-\frac {\frac {1}{3\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \]
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