Optimal. Leaf size=60 \[ \frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}}+\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3444, 2636, 2639} \[ \frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}}+\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2639
Rule 3444
Rubi steps
\begin {align*} \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx}{3 b}\\ &=\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}}+\frac {2 \int \sqrt {\cos (a+b x)} \, dx}{3 b}\\ &=\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)}+\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 54, normalized size = 0.90 \[ \frac {2 \left (-\sin (2 (a+b x))+2 \cos ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+b x\right )}{3 b^2 \cos ^{\frac {3}{2}}(a+b x)} \]
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 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x +a \right )}{\cos \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\sin \left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________