Optimal. Leaf size=45 \[ \frac {b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3098, 3133} \[ \frac {b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3098
Rule 3133
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx &=\frac {a x}{a^2+b^2}+\frac {b \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 41, normalized size = 0.91 \[ \frac {b \log (a \cos (c+d x)+b \sin (c+d x))+a (c+d x)}{d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 61, normalized size = 1.36 \[ \frac {2 \, a d x + b \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )}{2 \, {\left (a^{2} + b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 74, normalized size = 1.64 \[ \frac {\frac {2 \, b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}} + \frac {2 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 74, normalized size = 1.64 \[ \frac {b \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )}-\frac {b \ln \left (\tan ^{2}\left (d x +c \right )+1\right )}{2 d \left (a^{2}+b^{2}\right )}+\frac {a \arctan \left (\tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.60, size = 124, normalized size = 2.76 \[ \frac {\frac {2 \, a \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2} + b^{2}} + \frac {b \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} - \frac {b \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.16, size = 1069, normalized size = 23.76 \[ \frac {b\,\ln \left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{d\,\left (a^2+b^2\right )}-\frac {2\,a\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {\left (a^4-13\,a^2\,b^2+4\,b^4\right )\,\left (\frac {a^3\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^3}+\frac {a\,\left (96\,a\,b^2-32\,a^3+\frac {b\,\left (32\,a\,b^3+128\,a^3\,b-\frac {b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b\,\left (\frac {a\,\left (32\,a\,b^3+128\,a^3\,b-\frac {b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a\,b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}\right )}{{\left (a^4+5\,a^2\,b^2+4\,b^4\right )}^2}-\frac {6\,a\,b\,\left (a^2-2\,b^2\right )\,\left (32\,a\,b-\frac {b\,\left (96\,a\,b^2-32\,a^3+\frac {b\,\left (32\,a\,b^3+128\,a^3\,b-\frac {b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,\left (\frac {a\,\left (32\,a\,b^3+128\,a^3\,b-\frac {b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a\,b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}-\frac {a^2\,b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^3}\right )}{{\left (a^4+5\,a^2\,b^2+4\,b^4\right )}^2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{32\,a^2}+\frac {\left (a^4-13\,a^2\,b^2+4\,b^4\right )\,\left (\frac {a\,\left (32\,a^2\,b-\frac {b\,\left (64\,a^2\,b^2-32\,a^4+\frac {b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a^3\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}-\frac {b\,\left (\frac {a\,\left (64\,a^2\,b^2-32\,a^4+\frac {b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{32\,a^2\,{\left (a^4+5\,a^2\,b^2+4\,b^4\right )}^2}+\frac {3\,b\,\left (a^2-2\,b^2\right )\,\left (\frac {b\,\left (32\,a^2\,b-\frac {b\,\left (64\,a^2\,b^2-32\,a^4+\frac {b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,\left (\frac {a\,\left (64\,a^2\,b^2-32\,a^4+\frac {b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}+\frac {a^2\,b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{16\,a\,{\left (a^4+5\,a^2\,b^2+4\,b^4\right )}^2}\right )}{d\,\left (a^2+b^2\right )}-\frac {b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d\,\left (a^2+b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.75, size = 296, normalized size = 6.58 \[ \begin {cases} \frac {\tilde {\infty } x \cos {\relax (c )}}{\sin {\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {d x \sin {\left (c + d x \right )}}{2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} + \frac {i d x \cos {\left (c + d x \right )}}{2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} - \frac {\cos {\left (c + d x \right )}}{2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} & \text {for}\: a = - i b \\- \frac {d x \sin {\left (c + d x \right )}}{- 2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} - \frac {i d x \cos {\left (c + d x \right )}}{- 2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} - \frac {\cos {\left (c + d x \right )}}{- 2 i b d \sin {\left (c + d x \right )} + 2 b d \cos {\left (c + d x \right )}} & \text {for}\: a = i b \\\frac {x \cos {\relax (c )}}{a \cos {\relax (c )} + b \sin {\relax (c )}} & \text {for}\: d = 0 \\\frac {\log {\left (\sin {\left (c + d x \right )} \right )}}{b d} & \text {for}\: a = 0 \\\frac {a d x}{a^{2} d + b^{2} d} + \frac {b \log {\left (\cos {\left (c + d x \right )} + \frac {b \sin {\left (c + d x \right )}}{a} \right )}}{a^{2} d + b^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________