Optimal. Leaf size=62 \[ \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2626, 3771, 2639} \[ \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2626
Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=\frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx\\ &=\frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx\\ &=\frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\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)}}{b}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 54, normalized size = 0.87 \[ \frac {\sqrt {\csc (a+b x)} \left (\sin (a+b x) \tan (a+b x)+\sqrt {\sin (a+b x)} E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}}, x\right ) \]
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 \[ \int \frac {\sec \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 177, normalized size = 2.85 \[ \frac {\sqrt {\left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}\, \left (2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x +a \right )\right )+2\right )}{2 \sqrt {-\sin \left (b x +a \right ) \left (\sin \left (b x +a \right )-1\right ) \left (\sin \left (b x +a \right )+1\right )}\, \cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \frac {\sec \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (a+b\,x\right )}^2\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {\sec ^{2}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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