Optimal. Leaf size=38 \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3396}
\begin {gather*} \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\cos ^{\frac {3}{2}}(a+b x)}+x \sqrt {\cos (a+b x)}\right ) \, dx &=\int \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx+\int x \sqrt {\cos (a+b x)} \, dx\\ &=\frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 33, normalized size = 0.87 \begin {gather*} \frac {2 (2 \cos (a+b x)+b x \sin (a+b x))}{b^2 \sqrt {\cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (b x +a \right )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\left (b x +a \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x \left (\cos ^{2}{\left (a + b x \right )} + 1\right )}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 51, normalized size = 1.34 \begin {gather*} \frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\left (2\,\cos \left (2\,a+2\,b\,x\right )+b\,x\,\sin \left (2\,a+2\,b\,x\right )+2\right )}{b^2\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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