Integrand size = 23, antiderivative size = 429 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac {\left (176 a^2-167 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (96 a^2+179 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{40 d \sqrt {a+b \sin (c+d x)}}-\frac {5 b \left (12 a^2-b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.99 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2804, 3126, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {a \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{40 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {5 b \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2804
Rule 2884
Rule 2886
Rule 3081
Rule 3126
Rule 3128
Rule 3138
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {1}{4} \left (32 a^2-3 b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {1}{4} \left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2} \\ & = \frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {15}{8} b \left (12 a^2-b^2\right )-\frac {3}{4} a \left (8 a^2-5 b^2\right ) \sin (c+d x)-\frac {1}{8} b \left (208 a^2-25 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2} \\ & = -\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac {\int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {75}{16} a b \left (12 a^2-b^2\right )-\frac {3}{8} a^2 \left (40 a^2-71 b^2\right ) \sin (c+d x)-\frac {9}{16} a b \left (96 a^2-25 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{15 a^2} \\ & = -\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac {2 \int \frac {\csc (c+d x) \left (\frac {225}{32} a^2 b \left (12 a^2-b^2\right )-\frac {9}{16} a^3 \left (40 a^2-173 b^2\right ) \sin (c+d x)-\frac {9}{32} a^2 b \left (176 a^2-167 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{45 a^2} \\ & = -\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac {2 \int \frac {\csc (c+d x) \left (-\frac {225}{32} a^2 b^2 \left (12 a^2-b^2\right )-\frac {9}{32} a^3 b \left (96 a^2+179 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{45 a^2 b}-\frac {1}{80} \left (-176 a^2+167 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx \\ & = -\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}-\frac {1}{16} \left (5 b \left (12 a^2-b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {1}{80} \left (a \left (96 a^2+179 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (\left (-176 a^2+167 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}{80 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac {\left (176 a^2-167 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)}}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (5 b \left (12 a^2-b^2\right ) \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}{16 \sqrt {a+b \sin (c+d x)}}-\frac {\left (a \left (96 a^2+179 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}{80 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac {\left (176 a^2-167 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)}}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (96 a^2+179 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}}}{40 d \sqrt {a+b \sin (c+d x)}}-\frac {5 b \left (12 a^2-b^2\right ) \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}}}{8 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.73 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\frac {2 i \left (-176 a^2+167 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-\frac {8 a \left (40 a^2-173 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (424 a^2+117 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {4}{3} \sqrt {a+b \sin (c+d x)} \left (176 a b \cos (c+d x)+5 \cot (c+d x) \left (-32 a^2+33 b^2+26 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right )+24 b^2 \sin (2 (c+d x))\right )}{160 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1525\) vs. \(2(490)=980\).
Time = 8.23 (sec) , antiderivative size = 1526, normalized size of antiderivative = 3.56
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Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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