Optimal. Leaf size=43 \[ \frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0546732, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (1+\frac{a^2}{x^2}+\frac{2 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0228808, size = 38, normalized size = 0.88 \[ a^2 \left (\frac{\sin (c+d x)}{d}-\frac{\csc (c+d x)}{d}+\frac{2 \log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 46, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}}{d\sin \left ( dx+c \right ) }}+2\,{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.18064, size = 54, normalized size = 1.26 \begin{align*} \frac{2 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + a^{2} \sin \left (d x + c\right ) - \frac{a^{2}}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6657, size = 112, normalized size = 2.6 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{d \sin \left (d x + c\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 = 1.29898, size = 55, normalized size = 1.28 \begin{align*} \frac{2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a^{2} \sin \left (d x + c\right ) - \frac{a^{2}}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]