Optimal. Leaf size=28 \[ -\frac{2 B \sin (c+d x)}{3 d (a \cos (c+d x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0423808, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2749} \[ -\frac{2 B \sin (c+d x)}{3 d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2749
Rubi steps
\begin{align*} \int \frac{-\frac{5 B}{3}+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx &=-\frac{2 B \sin (c+d x)}{3 d (a+a \cos (c+d x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0671509, size = 28, normalized size = 1. \[ -\frac{2 B \sin (c+d x)}{3 d (a (\cos (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.828, size = 48, normalized size = 1.7 \begin{align*} -{\frac{B\sqrt{2}}{6\,{a}^{2}d}\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}{\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [B] time = 1.3452, size = 169, normalized size = 6.04 \begin{align*} -\frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a} B \sin \left (d x + c\right )}{3 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\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 [B] time = 3.49017, size = 80, normalized size = 2.86 \begin{align*} -\frac{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (\frac{\sqrt{2} B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3}} + \frac{\sqrt{2} B}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]