Optimal. Leaf size=33 \[ \frac {2 x}{b \sqrt {\cos (a+b x)}}-\frac {4 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3444, 2641} \[ \frac {2 x}{b \sqrt {\cos (a+b x)}}-\frac {4 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 3444
Rubi steps
\begin {align*} \int \frac {x \sin (a+b x)}{\cos ^{\frac {3}{2}}(a+b x)} \, dx &=\frac {2 x}{b \sqrt {\cos (a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\cos (a+b x)}}-\frac {4 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 33, normalized size = 1.00 \[ \frac {2 x}{b \sqrt {\cos (a+b x)}}-\frac {4 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2} \]
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 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x +a \right )}{\cos \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\sin \left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin {\left (a + b x \right )}}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________