Optimal. Leaf size=92 \[ \frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{15 b} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2628, 3771, 2639} \[ \frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{15 b} \]
Antiderivative was successfully verified.
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Rule 2628
Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2}{3} \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx\\ &=\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4}{15} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx\\ &=\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {1}{15} \left (4 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx\\ &=\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{15 b}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 63, normalized size = 0.68 \[ \frac {39 \cos (a+b x)+5 \cos (3 (a+b x))-\frac {48 E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )}{\sin ^{\frac {3}{2}}(a+b x)}}{90 b \csc ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}}, x\right ) \]
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 \[ \int \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 152, normalized size = 1.65 \[ \frac {-\frac {2 \left (\cos ^{6}\left (b x +a \right )\right )}{9}-\frac {8 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {2 \left (\cos ^{4}\left (b x +a \right )\right )}{45}+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{15}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x\right )}^4}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {\cos ^{4}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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