Optimal. Leaf size=18 \[ \frac {\log \left (\sin ^2(c+d x)+1\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 260
Rule 4334
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 = 0.11, size = 20, normalized size = 1.11 \[ \frac {\log (3-\cos (2 (c+d x)))}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 18, normalized size = 1.00 \[ \frac {\log \left (-\cos \left (d x + c\right )^{2} + 2\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 17, normalized size = 0.94 \[ \frac {\ln \left (\cos ^{2}\left (d x +c \right )-2\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 16, normalized size = 0.89 \[ \frac {\ln \left ({\sin \left (c+d\,x\right )}^2+1\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________