Optimal. Leaf size=430 \[ -\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}+\frac {a \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 a \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \sin (c+d x)}}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d} \]
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Rubi [A] time = 1.42, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2893, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (3 a^2 b^2+8 a^4-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{28 b d \sqrt {a+b \sin (c+d x)}}+\frac {a \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 a \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \sin (c+d x)}}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 2893
Rule 3002
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac {3}{4} \left (4 a^2-5 b^2\right )+\frac {5}{2} a b \sin (c+d x)-\frac {1}{4} \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {21}{8} a \left (4 a^2-5 b^2\right )+\frac {57}{4} a^2 b \sin (c+d x)-\frac {5}{8} a \left (8 a^2-35 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{7 a^2}\\ &=-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {2 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {105}{16} a^2 \left (4 a^2-5 b^2\right )+\frac {435}{8} a^3 b \sin (c+d x)-\frac {15}{16} a^2 \left (8 a^2-73 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{35 a^2}\\ &=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {4 \int \frac {\csc (c+d x) \left (\frac {315}{32} a^3 \left (4 a^2-5 b^2\right )+\frac {15}{16} a^2 b \left (125 a^2-16 b^2\right ) \sin (c+d x)-\frac {15}{32} a^3 \left (8 a^2-247 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 a^2}\\ &=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {4 \int \frac {\csc (c+d x) \left (-\frac {315}{32} a^3 b \left (4 a^2-5 b^2\right )-\frac {15}{32} a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 a^2 b}+\frac {\left (a \left (8 a^2-247 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{56 b}\\ &=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}-\frac {1}{8} \left (3 a \left (4 a^2-5 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{56 b}+\frac {\left (a \left (8 a^2-247 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}{56 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 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)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 a \left (4 a^2-5 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}{8 \sqrt {a+b \sin (c+d x)}}-\frac {\left (\left (8 a^4+3 a^2 b^2-32 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}{56 b \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 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)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\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}}}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 a \left (4 a^2-5 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}}}{4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 5.54, size = 460, normalized size = 1.07 \[ \frac {\frac {8 b \left (125 a^2-16 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (160 a^2+37 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+4 \sqrt {a+b \sin (c+d x)} \left (\left (22 b^2-24 a^2\right ) \cos (c+d x)-12 a b \sin (2 (c+d x))-7 a \cot (c+d x) (2 a \csc (c+d x)+9 b)+2 b^2 \cos (3 (c+d x))\right )+\frac {2 i \left (247 b^2-8 a^2\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {\frac {b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{b^2 \sqrt {-\frac {1}{a+b}}}}{112 d} \]
Antiderivative was successfully verified.
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fricas [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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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 = 2.24, size = 1520, normalized size = 3.53 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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