Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2621, 329, 212, 206, 203} \[ \frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 2621
Rubi steps
\begin {align*} \int \frac {\sec (a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b}\\ &=\frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 1.61 \[ -\frac {\sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \left (\tan ^{-1}\left (\sqrt {\sin (a+b x)}\right )-\tanh ^{-1}\left (\sqrt {\sin (a+b x)}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 97, normalized size = 3.13 \[ -\frac {2 \, \arctan \left (\frac {\sin \left (b x + a\right ) - 1}{2 \, \sqrt {\sin \left (b x + a\right )}}\right ) - \log \left (\frac {\cos \left (b x + a\right )^{2} + \frac {4 \, {\left (\cos \left (b x + a\right )^{2} - \sin \left (b x + a\right ) - 1\right )}}{\sqrt {\sin \left (b x + a\right )}} - 6 \, \sin \left (b x + a\right ) - 2}{\cos \left (b x + a\right )^{2} + 2 \, \sin \left (b x + a\right ) - 2}\right )}{4 \, b} \]
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 \[ \int \frac {\sec \left (b x + a\right )}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 48, normalized size = 1.55 \[ -\frac {\ln \left (\sqrt {\sin }\left (b x +a \right )-1\right )}{2 b}+\frac {\ln \left (\sqrt {\sin }\left (b x +a \right )+1\right )}{2 b}-\frac {\arctan \left (\sqrt {\sin }\left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 41, normalized size = 1.32 \[ \frac {2 \, \arctan \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}}\right ) + \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} + 1\right ) - \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} - 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\cos \left (a+b\,x\right )\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {\sec {\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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