Optimal. Leaf size=18 \[ \frac{\log \left (\sin ^2(c+d x)+1\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0311779, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4334, 260} \[ \frac{\log \left (\sin ^2(c+d x)+1\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4334
Rule 260
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\log \left (1+\sin ^2(c+d x)\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.107006, size = 20, normalized size = 1.11 \[ \frac{\log (3-\cos (2 (c+d x)))}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 17, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2 \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.67404, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.492568, size = 43, normalized size = 2.39 \begin{align*} \frac{\log \left (-\cos \left (d x + c\right )^{2} + 2\right )}{2 \, d} \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*} \int \frac{\cos{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + \csc{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15153, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]