Optimal. Leaf size=29 \[ \frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0523097, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2633} \[ \frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3175
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^7(x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \cos ^5(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )}{a}\\ &=\frac{\sin (x)}{a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin ^5(x)}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0030325, size = 27, normalized size = 0.93 \[ \frac{\frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 20, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+\sin \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0433, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.53558, size = 61, normalized size = 2.1 \begin{align*} \frac{{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right )}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 35.1572, size = 311, normalized size = 10.72 \begin{align*} \frac{30 \tan ^{9}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{40 \tan ^{7}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{116 \tan ^{5}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{40 \tan ^{3}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{30 \tan{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14075, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]