Optimal. Leaf size=55 \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\text {Int}\left (x \sqrt {\cos (a+b x)},x\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\cos ^{\frac {3}{2}}(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)}}-\int x \sqrt {\cos (a+b x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (b x +a \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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 [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\cos \left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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