Optimal. Leaf size=412 \[ \frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}-\frac {b \left (188 a^2-105 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{192 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (48 a^4-72 a^2 b^2+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{64 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (68 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{192 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d} \]
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Rubi [A] time = 1.24, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2893, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {b \left (68 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{192 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (188 a^2-105 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{192 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (-72 a^2 b^2+48 a^4+35 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{64 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d} \]
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 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (\frac {5}{4} \left (12 a^2-7 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {3}{4} \left (16 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{12 a^2}\\ &=\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {1}{8} b \left (188 a^2-105 b^2\right )-\frac {1}{4} a \left (36 a^2-7 b^2\right ) \sin (c+d x)+\frac {5}{8} b \left (12 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{24 a^3}\\ &=-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {3}{16} \left (48 a^4-72 a^2 b^2+35 b^4\right )+\frac {5}{8} a b \left (12 a^2-7 b^2\right ) \sin (c+d x)+\frac {1}{16} b^2 \left (188 a^2-105 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{24 a^4}\\ &=-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {3}{16} b \left (48 a^4-72 a^2 b^2+35 b^4\right )+\frac {1}{16} a b^2 \left (68 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{24 a^4 b}-\frac {\left (b \left (188 a^2-105 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{384 a^4}\\ &=-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}+\frac {\left (b \left (68 a^2-35 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{384 a^3}+\frac {\left (48 a^4-72 a^2 b^2+35 b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{128 a^4}-\frac {\left (b \left (188 a^2-105 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}{384 a^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}-\frac {b \left (188 a^2-105 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)}}{192 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (b \left (68 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}{384 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (48 a^4-72 a^2 b^2+35 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}{128 a^4 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{192 a^4 d}+\frac {5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{96 a^3 d}+\frac {7 b \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 a d}-\frac {b \left (188 a^2-105 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)}}{192 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (68 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}}}{192 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (48 a^4-72 a^2 b^2+35 b^4\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}}}{64 a^4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.70, size = 647, normalized size = 1.57 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {7 b \cot (c+d x) \csc ^2(c+d x)}{24 a^2}+\frac {\csc (c+d x) \left (105 b^3 \cos (c+d x)-188 a^2 b \cos (c+d x)\right )}{192 a^4}+\frac {5 \csc ^2(c+d x) \left (12 a^2 \cos (c+d x)-7 b^2 \cos (c+d x)\right )}{96 a^3}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a}\right )}{d}+\frac {-\frac {2 \left (140 a b^3-240 a^3 b\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (188 a^2 b^2-105 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt {-\frac {a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac {2 \left (288 a^4-620 a^2 b^2+315 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}}{768 a^4 d} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.34, size = 1761, normalized size = 4.27 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2}{{\sin \left (c+d\,x\right )}^5\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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