Optimal. Leaf size=34 \[ -\frac{\cos (e+f x)}{f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.045294, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3186, 191} \[ -\frac{\cos (e+f x)}{f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3186
Rule 191
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x)}{(a+b) f \sqrt{a+b-b \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.114126, size = 41, normalized size = 1.21 \[ -\frac{\sqrt{2} \cos (e+f x)}{f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.839, size = 31, normalized size = 0.9 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{ \left ( a+b \right ) f}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.945119, size = 43, normalized size = 1.26 \begin{align*} -\frac{\cos \left (f x + e\right )}{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55985, size = 136, normalized size = 4. \begin{align*} \frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \cos \left (f x + e\right )}{{\left (a b + b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{2} + 2 \, a b + b^{2}\right )} f} \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 [A] time = 1.3584, size = 72, normalized size = 2.12 \begin{align*} \frac{\sqrt{-{\left (\cos \left (f x + e\right )^{2} - 1\right )} b + a} \cos \left (f x + e\right )}{{\left ({\left (\cos \left (f x + e\right )^{2} - 1\right )} b - a\right )}{\left (a + b\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]