Optimal. Leaf size=92 \[ \frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {8 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{7 b} \]
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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, 2641} \[ \frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {8 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{7 b} \]
Antiderivative was successfully verified.
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Rule 2628
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx &=\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {6}{7} \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx\\ &=\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {4}{7} \int \sqrt {\csc (a+b x)} \, dx\\ &=\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {1}{7} \left (4 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {8 \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 63, normalized size = 0.68 \[ \frac {\sqrt {\csc (a+b x)} \left (10 \sin (2 (a+b x))+\sin (4 (a+b x))-32 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{28 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )}, x\right ) \]
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 \[ \int \cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 100, normalized size = 1.09 \[ \frac {\frac {2 \left (\sin ^{5}\left (b x +a \right )\right )}{7}-\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{7}+\frac {6 \sin \left (b x +a \right )}{7}+\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{7}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )}\,{d x} \]
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 \[ \int {\cos \left (a+b\,x\right )}^4\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]
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 \[ \int \cos ^{4}{\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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