Integrand size = 31, antiderivative size = 482 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+492 a^2 b^2-5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (80 a^4-40 a^2 b^2+b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^2 d \sqrt {a+b \sin (c+d x)}} \]
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Time = 1.11 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2972, 3126, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {\left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{640 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{640 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {3 b \left (80 a^4-40 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2972
Rule 3081
Rule 3126
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {3}{4} \left (32 a^2-b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2+3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2} \\ & = \frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {15}{8} b \left (36 a^2-b^2\right )-\frac {3}{4} a \left (16 a^2-5 b^2\right ) \sin (c+d x)-\frac {3}{8} b \left (192 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2} \\ & = \frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\int \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (-\frac {3}{16} \left (128 a^4-580 a^2 b^2+15 b^4\right )-\frac {3}{8} a b \left (268 a^2-5 b^2\right ) \sin (c+d x)-\frac {9}{16} b^2 \left (316 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2} \\ & = -\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {45}{32} b \left (80 a^4-40 a^2 b^2+b^4\right )-\frac {3}{16} a b^2 \left (1484 a^2+5 b^2\right ) \sin (c+d x)+\frac {3}{32} b \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^2} \\ & = -\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {45}{32} b^2 \left (80 a^4-40 a^2 b^2+b^4\right )+\frac {3}{32} a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^2 b}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{1280 a^2} \\ & = -\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}+\frac {\left (128 a^4+492 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{1280 a}+\frac {\left (3 b \left (80 a^4-40 a^2 b^2+b^4\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{256 a^2}-\frac {\left (\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{1280 a^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \\ & = -\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\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)}}{640 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (128 a^4+492 a^2 b^2-5 b^4\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}{1280 a \sqrt {a+b \sin (c+d x)}}+\frac {\left (3 b \left (80 a^4-40 a^2 b^2+b^4\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}{256 a^2 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\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)}}{640 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+492 a^2 b^2-5 b^4\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}}}{640 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (80 a^4-40 a^2 b^2+b^4\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}}}{128 a^2 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.33 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.13 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-4 \cot (c+d x) \left (128 a^4-1196 a^2 b^2-15 b^4+\left (-872 a^3 b+10 a b^3\right ) \csc (c+d x)-8 a^2 \left (32 a^2-31 b^2\right ) \csc ^2(c+d x)+336 a^3 b \csc ^3(c+d x)+128 a^4 \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}+b \left (-\frac {2 i \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {8 a b \left (1484 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (2272 a^4+1276 a^2 b^2+45 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{2560 a^2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2074\) vs. \(2(543)=1086\).
Time = 50.17 (sec) , antiderivative size = 2075, normalized size of antiderivative = 4.30
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\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]
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