Optimal. Leaf size=429 \[ -\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {a \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{40 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {5 b \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d} \]
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Rubi [A] time = 1.36, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2725, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {a \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{40 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {5 b \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2725
Rule 2805
Rule 2807
Rule 3002
Rule 3047
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\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 ) F\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 ) \Pi \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*}
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Mathematica [C] time = 3.65, size = 466, normalized size = 1.09 \[ \frac {-\frac {8 a \left (40 a^2-173 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (424 a^2+117 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {4}{3} \sqrt {a+b \sin (c+d x)} \left (5 \cot (c+d x) \left (8 a^2 \csc ^2(c+d x)-32 a^2+26 a b \csc (c+d x)+33 b^2\right )+176 a b \cos (c+d x)+24 b^2 \sin (2 (c+d x))\right )+\frac {2 i \left (167 b^2-176 a^2\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}}{160 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 124.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, a b \cot \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \cot \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.06, size = 1526, normalized size = 3.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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