Optimal. Leaf size=38 \[ \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2636, 2639} \[ \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx &=\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\int \sqrt {\cos (a+b x)} \, dx\\ &=-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}+\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 38, normalized size = 1.00 \[ \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 101, normalized size = 2.66 \[ -\frac {2 \left (\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{\sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 42, normalized size = 1.11 \[ \frac {2\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,\sqrt {\cos \left (a+b\,x\right )}\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
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 \[ \int \frac {1}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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