Optimal. Leaf size=48 \[ \frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.115148, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2836, 12, 77} \[ \frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2836
Rule 12
Rule 77
Rubi steps
\begin{align*} \int \csc (c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{a^2 (-a+x)}{(-a-x) x^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{-a+x}{(-a-x) x^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{2}{a x}+\frac{2}{a (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}+\frac{a^2 \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0755605, size = 36, normalized size = 0.75 \[ \frac{a^2 \left (\sec (c+d x)+4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 \log (\cos (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 32, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}\sec \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.975907, size = 58, normalized size = 1.21 \begin{align*} \frac{2 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - 2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac{a^{2}}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7431, size = 155, normalized size = 3.23 \begin{align*} -\frac{2 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 2 \, a^{2} \cos \left (d x + c\right ) \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - a^{2}}{d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int 2 \csc{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \csc{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.39435, size = 155, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{2 \, a^{2} + \frac{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]