Optimal. Leaf size=22 \[ \frac {\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {4383} \[ \frac {\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4383
Rubi steps
\begin {align*} \int \frac {a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{b \sec (c+d x)+a \sin (c+d x)} \, dx &=\frac {\log (b \sec (c+d x)+a \sin (c+d x))}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.46, size = 29, normalized size = 1.32 \[ \frac {\log (a \sin (2 (c+d x))+2 b)-\log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.13, size = 33, normalized size = 1.50 \[ \frac {\log \left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right ) - \log \left (-\cos \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.52, size = 23, normalized size = 1.05 \[ \frac {\ln \left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 22, normalized size = 1.00 \[ \frac {\log \left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.86, size = 133, normalized size = 6.05 \[ \frac {\mathrm {atan}\left (\frac {-\cos \left (c+d\,x\right )\,a^6+8\,\cos \left (c+d\,x\right )\,a^4\,b^2-16\,\cos \left (c+d\,x\right )\,a^2\,b^4+\frac {\sin \left (2\,c+2\,d\,x\right )\,a\,b^5}{2}+b^6}{1{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^6-8{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^4\,b^2+16{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^2\,b^4+\frac {1{}\mathrm {i}\,\sin \left (2\,c+2\,d\,x\right )\,a\,b^5}{2}+b^6\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.48, size = 63, normalized size = 2.86 \[ \begin {cases} x \tan {\relax (c )} & \text {for}\: a = 0 \wedge d = 0 \\\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: a = 0 \\\frac {x \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right )}{a \sin {\relax (c )} + b \sec {\relax (c )}} & \text {for}\: d = 0 \\\frac {\log {\left (\sin {\left (c + d x \right )} + \frac {b \sec {\left (c + d x \right )}}{a} \right )}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________