Optimal. Leaf size=79 \[ \frac{4 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0589234, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4303, 4304, 4291} \[ \frac{4 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4303
Rule 4304
Rule 4291
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx &=-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{4}{5} \int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{4 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{8}{15} \int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{4 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.120671, size = 52, normalized size = 0.66 \[ -\frac{\sqrt{\sin (2 (a+b x))} \left (3 \csc ^3(a+b x)+27 \csc (a+b x)-5 \tan (a+b x) \sec (a+b x)\right )}{120 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 89.464, size = 481, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.514449, size = 269, normalized size = 3.41 \begin{align*} -\frac{\sqrt{2}{\left (32 \, \cos \left (b x + a\right )^{4} - 40 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \,{\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{120 \,{\left (b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]