Optimal. Leaf size=62 \[ \frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2621, 288, 329, 212, 206, 203} \[ \frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 288
Rule 329
Rule 2621
Rubi steps
\begin {align*} \int \frac {\sec ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^{3/2}}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\csc (a+b x)\right )}{4 b}\\ &=\frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\csc (a+b x)}\right )}{2 b}\\ &=\frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{4 b}\\ &=\frac {\tan ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b}+\frac {\tanh ^{-1}\left (\sqrt {\csc (a+b x)}\right )}{4 b}+\frac {\sec ^2(a+b x)}{2 b \csc ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 33, normalized size = 0.53 \[ \frac {2 \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};\sin ^2(a+b x)\right )}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 141, normalized size = 2.27 \[ -\frac {2 \, \arctan \left (\frac {\sin \left (b x + a\right ) - 1}{2 \, \sqrt {\sin \left (b x + a\right )}}\right ) \cos \left (b x + a\right )^{2} - \cos \left (b x + a\right )^{2} \log \left (\frac {\cos \left (b x + a\right )^{2} + \frac {4 \, {\left (\cos \left (b x + a\right )^{2} - \sin \left (b x + a\right ) - 1\right )}}{\sqrt {\sin \left (b x + a\right )}} - 6 \, \sin \left (b x + a\right ) - 2}{\cos \left (b x + a\right )^{2} + 2 \, \sin \left (b x + a\right ) - 2}\right ) + \frac {8 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )}}{\sqrt {\sin \left (b x + a\right )}}}{16 \, b \cos \left (b x + a\right )^{2}} \]
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 \[ \int \frac {\sec \left (b x + a\right )^{3}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 71, normalized size = 1.15 \[ \frac {-\left (\ln \left (\sqrt {\sin }\left (b x +a \right )-1\right )-\ln \left (\sqrt {\sin }\left (b x +a \right )+1\right )+2 \arctan \left (\sqrt {\sin }\left (b x +a \right )\right )\right ) \left (\cos ^{2}\left (b x +a \right )\right )+4 \left (\sin ^{\frac {3}{2}}\left (b x +a \right )\right )}{8 \cos \left (b x +a \right )^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 63, normalized size = 1.02 \[ \frac {\frac {4}{{\left (\frac {1}{\sin \left (b x + a\right )^{2}} - 1\right )} \sqrt {\sin \left (b x + a\right )}} + 2 \, \arctan \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}}\right ) + \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} + 1\right ) - \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} - 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (a+b\,x\right )}^3\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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