Optimal. Leaf size=63 \[ \frac{a^4 \cos (c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0691645, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2670, 2650, 2648} \[ \frac{a^4 \cos (c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2670
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac{1}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{1}{3} a^3 \int \frac{1}{a-a \sin (c+d x)} \, dx\\ &=\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{a^3 \cos (c+d x)}{3 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0086958, size = 58, normalized size = 0.92 \[ -\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.065, size = 63, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,{a}^{2}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.953012, size = 70, normalized size = 1.11 \begin{align*} \frac{a^{2} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + \frac{2 \, a^{2}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.67741, size = 235, normalized size = 3.73 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2} -{\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, 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 [A] time = 1.2081, size = 73, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a^{2}\right )}}{3 \, d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]