Optimal. Leaf size=416 \[ \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}+\frac {5 \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{24 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 1.15, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2724, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{24 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a^4 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2724
Rule 2805
Rule 2807
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {2 \int \frac {\csc ^3(c+d x) \left (\frac {1}{4} \left (24 a^2-35 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {3}{4} \left (4 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^2 b}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {5}{8} b \left (16 a^2-21 b^2\right )+\frac {7}{4} a b^2 \sin (c+d x)+\frac {1}{8} b \left (24 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^3 b}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {\int \frac {\csc (c+d x) \left (\frac {3}{16} b^2 \left (36 a^2-35 b^2\right )+\frac {1}{8} a b \left (24 a^2-35 b^2\right ) \sin (c+d x)+\frac {5}{16} b^2 \left (16 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^4 b}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{16} b^3 \left (36 a^2-35 b^2\right )+\frac {1}{16} a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^4 b^2}+\frac {\left (5 \left (16 a^2-21 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{48 a^4}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}-\frac {\left (32 a^2-35 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{48 a^3}+\frac {\left (b \left (36 a^2-35 b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{16 a^4}+\frac {\left (5 \left (16 a^2-21 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}{48 a^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 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)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (\left (32 a^2-35 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}{48 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (b \left (36 a^2-35 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 a^4 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 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)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-35 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}}}{24 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (36 a^2-35 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 a^4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 5.85, size = 468, normalized size = 1.12 \[ \frac {-\frac {4 \left (\left (105 b^3-80 a^2 b\right ) \cos (c+d x)+a \cot (c+d x) \left (8 a^2 \csc ^2(c+d x)-32 a^2-14 a b \csc (c+d x)+35 b^2\right )\right )}{a^4 \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {8 a \left (24 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (315 b^2-296 a^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {10 i \left (21 b^2-16 a^2\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}}{a^4}}{96 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.18, size = 1496, normalized size = 3.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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