Optimal. Leaf size=38 \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, 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 = {3315} \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3315
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.40, size = 33, normalized size = 0.87 \[ \frac {2 (b x \sin (a+b x)+2 \cos (a+b x))}{b^2 \sqrt {\cos (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cos \left (b x + a\right )} + \frac {x}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.18, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cos \left (b x + a\right )} + \frac {x}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.66, size = 51, normalized size = 1.34 \[ \frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\left (2\,\cos \left (2\,a+2\,b\,x\right )+b\,x\,\sin \left (2\,a+2\,b\,x\right )+2\right )}{b^2\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\cos ^{2}{\left (a + b x \right )} + 1\right )}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________