Integrand size = 31, antiderivative size = 285 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\left (4 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 a b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 a^2-7 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {b \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.62 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2973, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {\left (4 a^2-7 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{3 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 a b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {b \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{a d \sqrt {a+b \sin (c+d x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2973
Rule 3081
Rule 3138
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {2 \int \frac {\csc (c+d x) \left (\frac {3 b^2}{4}+\frac {5}{2} a b \sin (c+d x)+\frac {1}{4} \left (4 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a b} \\ & = -\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}+\frac {2 \int \frac {\csc (c+d x) \left (-\frac {3 b^3}{4}+\frac {1}{4} a \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a b^2}-\frac {\left (4 a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{6 a b^2} \\ & = -\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}+\frac {1}{6} \left (-7+\frac {4 a^2}{b^2}\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {b \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 a}-\frac {\left (\left (4 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{6 a b^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\left (4 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 a b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (-7+\frac {4 a^2}{b^2}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{6 \sqrt {a+b \sin (c+d x)}}-\frac {\left (b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{2 a \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {\cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\left (4 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 a b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (7-\frac {4 a^2}{b^2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 d \sqrt {a+b \sin (c+d x)}}-\frac {b \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.63 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {\frac {2 i \left (4 a^2+3 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a^2 b^3 \sqrt {-\frac {1}{a+b}}}-\frac {4 \cot (c+d x) (3 b+2 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{a b}+\frac {40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (4 a^2+9 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a b \sqrt {a+b \sin (c+d x)}}}{12 d} \]
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Time = 1.08 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.47
method | result | size |
default | \(\frac {\sqrt {-\left (-b \sin \left (d x +c \right )-a \right ) \left (\cos ^{2}\left (d x +c \right )\right )}\, \left (2 a^{2} b^{3} \left (\cos ^{4}\left (d x +c \right )\right )+\left (-2 a^{3} b^{2}-3 a \,b^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5 a^{2} b^{3} \left (\cos ^{2}\left (d x +c \right )\right )+\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \left (3 \Pi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a b -b^{2}}{a b}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-3 \Pi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a b -b^{2}}{a b}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}+4 E\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}-E\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-3 E\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-4 F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b -6 F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+7 F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}+3 F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}\right ) \sin \left (d x +c \right )\right )}{3 b^{3} \sqrt {b \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+a \left (\cos ^{2}\left (d x +c \right )\right )}\, a^{2} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(704\) |
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\[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\mathrm {cot}\left (c+d\,x\right )}^2}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
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