Optimal. Leaf size=67 \[ \frac{\sin (x) \cos (x)}{a \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^3(x)}{5 a \sqrt{a \cos ^4(x)}}+\frac{2 \sin ^2(x) \tan (x)}{3 a \sqrt{a \cos ^4(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0214227, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3767} \[ \frac{\sin (x) \cos (x)}{a \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^3(x)}{5 a \sqrt{a \cos ^4(x)}}+\frac{2 \sin ^2(x) \tan (x)}{3 a \sqrt{a \cos ^4(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cos ^4(x)\right )^{3/2}} \, dx &=\frac{\cos ^2(x) \int \sec ^6(x) \, dx}{a \sqrt{a \cos ^4(x)}}\\ &=-\frac{\cos ^2(x) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )}{a \sqrt{a \cos ^4(x)}}\\ &=\frac{\cos (x) \sin (x)}{a \sqrt{a \cos ^4(x)}}+\frac{2 \sin ^2(x) \tan (x)}{3 a \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^3(x)}{5 a \sqrt{a \cos ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0275874, size = 30, normalized size = 0.45 \[ \frac{\sin (x) \cos (x) (6 \cos (2 x)+\cos (4 x)+8)}{15 \left (a \cos ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.114, size = 29, normalized size = 0.4 \begin{align*}{\frac{\sin \left ( x \right ) \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{4}+4\, \left ( \cos \left ( x \right ) \right ) ^{2}+3 \right ) \cos \left ( x \right ) }{15} \left ( a \left ( \cos \left ( x \right ) \right ) ^{4} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.8994, size = 30, normalized size = 0.45 \begin{align*} \frac{3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.04243, size = 101, normalized size = 1.51 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}}{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, a^{2} \cos \left (x\right )^{7}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]