Integrand size = 23, antiderivative size = 96 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {(a+b)^{5/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {(a+b)^2 \cot (c+d x)}{a^3 d}+\frac {(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d} \]
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Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3274, 331, 211} \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {(a+b)^{5/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {(a+b)^2 \cot (c+d x)}{a^3 d}+\frac {(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d} \]
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Rule 211
Rule 331
Rule 3274
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^6 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\cot ^5(c+d x)}{5 a d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{a^2 d} \\ & = -\frac {(a+b)^2 \cot (c+d x)}{a^3 d}+\frac {(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {(a+b)^3 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a^3 d} \\ & = -\frac {(a+b)^{5/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {(a+b)^2 \cot (c+d x)}{a^3 d}+\frac {(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {-15 (a+b)^{5/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )-\sqrt {a} \cot (c+d x) \left (23 a^2+35 a b+15 b^2-a (11 a+5 b) \csc ^2(c+d x)+3 a^2 \csc ^4(c+d x)\right )}{15 a^{7/2} d} \]
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Time = 5.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {\left (-a^{3}-3 a^{2} b -3 a \,b^{2}-b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{3} \sqrt {a \left (a +b \right )}}-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {-a -b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+2 a b +b^{2}}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(115\) |
default | \(\frac {\frac {\left (-a^{3}-3 a^{2} b -3 a \,b^{2}-b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{3} \sqrt {a \left (a +b \right )}}-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {-a -b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+2 a b +b^{2}}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(115\) |
risch | \(-\frac {2 i \left (45 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+45 a b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-90 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-150 a b \,{\mathrm e}^{6 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+140 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+200 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+90 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-70 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-130 a b \,{\mathrm e}^{2 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+23 a^{2}+35 a b +15 b^{2}\right )}{15 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 a^{2} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b}{a^{3} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b^{2}}{2 a^{4} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 a^{2} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b}{a^{3} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b^{2}}{2 a^{4} d}\) | \(501\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (84) = 168\).
Time = 0.35 (sec) , antiderivative size = 576, normalized size of antiderivative = 6.00 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [-\frac {4 \, {\left (23 \, a^{2} + 35 \, a b + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 20 \, {\left (7 \, a^{2} + 13 \, a b + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\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 ) + 60 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )}, -\frac {2 \, {\left (23 \, a^{2} + 35 \, a b + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (7 \, a^{2} + 13 \, a b + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\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 ) + 30 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\cot ^{6}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{3}} + \frac {15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{4} - 5 \, {\left (a^{2} + a b\right )} \tan \left (d x + c\right )^{2} + 3 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{5}}}{15 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (84) = 168\).
Time = 0.54 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{\sqrt {a^{2} + a b} a^{3}} + \frac {15 \, a^{2} \tan \left (d x + c\right )^{4} + 30 \, a b \tan \left (d x + c\right )^{4} + 15 \, b^{2} \tan \left (d x + c\right )^{4} - 5 \, a^{2} \tan \left (d x + c\right )^{2} - 5 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{5}}}{15 \, d} \]
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Time = 13.96 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {1}{5\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{3\,a^2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (a+b\right )}^2}{a^3}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5}-\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,{\left (a+b\right )}^{5/2}}{a^{7/2}\,d} \]
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