Optimal. Leaf size=92 \[ \frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {5 \sec (a+b x)}{6 b \sqrt {\csc (a+b x)}}+\frac {5 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{6 b} \]
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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 = {2626, 3771, 2641} \[ \frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {5 \sec (a+b x)}{6 b \sqrt {\csc (a+b x)}}+\frac {5 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2626
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {\csc (a+b x)} \sec ^4(a+b x) \, dx &=\frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {5}{6} \int \sqrt {\csc (a+b x)} \sec ^2(a+b x) \, dx\\ &=\frac {5 \sec (a+b x)}{6 b \sqrt {\csc (a+b x)}}+\frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {5}{12} \int \sqrt {\csc (a+b x)} \, dx\\ &=\frac {5 \sec (a+b x)}{6 b \sqrt {\csc (a+b x)}}+\frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {1}{12} \left (5 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {5 \sec (a+b x)}{6 b \sqrt {\csc (a+b x)}}+\frac {\sec ^3(a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {5 \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)}}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 64, normalized size = 0.70 \[ \frac {\sqrt {\csc (a+b x)} \left (\tan (a+b x) \left (2 \sec ^2(a+b x)+5\right )-5 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{6 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}, x\right ) \]
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 \[ \int \sqrt {\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 168, normalized size = 1.83 \[ -\frac {\sqrt {\left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}\, \left (5 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (b x +a \right )\right )+10 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )+4 \sin \left (b x +a \right )\right )}{12 \left (\sin \left (b x +a \right )+1\right ) \sqrt {-\sin \left (b x +a \right ) \left (\sin \left (b x +a \right )-1\right ) \left (\sin \left (b x +a \right )+1\right )}\, \left (\sin \left (b x +a \right )-1\right ) \cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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 = -0.01 \[ \int \frac {\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}}{{\cos \left (a+b\,x\right )}^4} \,d x \]
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 \[ \int \sqrt {\csc {\left (a + b x \right )}} \sec ^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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