Optimal. Leaf size=33 \[ \frac {2}{b \sqrt {\csc (a+b x)}}-\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2621, 14} \[ \frac {2}{b \sqrt {\csc (a+b x)}}-\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2621
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{x^{7/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^{7/2}}+\frac {1}{x^{3/2}}\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {2}{b \sqrt {\csc (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 27, normalized size = 0.82 \[ \frac {\cos (2 (a+b x))+9}{5 b \sqrt {\csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 23, normalized size = 0.70 \[ \frac {2 \, {\left (\cos \left (b x + a\right )^{2} + 4\right )} \sqrt {\sin \left (b x + a\right )}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.09, size = 24, normalized size = 0.73 \[ -\frac {2 \, {\left (\sin \left (b x + a\right )^{\frac {5}{2}} - 5 \, \sqrt {\sin \left (b x + a\right )}\right )}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 26, normalized size = 0.79 \[ \frac {-\frac {2 \left (\sin ^{\frac {5}{2}}\left (b x +a \right )\right )}{5}+2 \left (\sqrt {\sin }\left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.70, size = 25, normalized size = 0.76 \[ \frac {2 \, {\left (\frac {5}{\sin \left (b x + a\right )^{2}} - 1\right )} \sin \left (b x + a\right )^{\frac {5}{2}}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\cos \left (a+b\,x\right )}^3\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]
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]
________________________________________________________________________________________