Optimal. Leaf size=75 \[ -\frac {a \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)}+\frac {\log (\cos (c+d x)+1)}{2 d (a-b)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4397, 2668, 706, 31, 633} \[ -\frac {a \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)}+\frac {\log (\cos (c+d x)+1)}{2 d (a-b)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 633
Rule 706
Rule 2668
Rule 4397
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx &=\int \frac {\csc (c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{(b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cos (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {a \operatorname {Subst}\left (\int \frac {-b+x}{a^2-x^2} \, dx,x,a \cos (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac {a \log (b+a \cos (c+d x))}{\left (a^2-b^2\right ) d}-\frac {\operatorname {Subst}\left (\int \frac {1}{-a-x} \, dx,x,a \cos (c+d x)\right )}{2 (a-b) d}-\frac {\operatorname {Subst}\left (\int \frac {1}{a-x} \, dx,x,a \cos (c+d x)\right )}{2 (a+b) d}\\ &=\frac {\log (1-\cos (c+d x))}{2 (a+b) d}+\frac {\log (1+\cos (c+d x))}{2 (a-b) d}-\frac {a \log (b+a \cos (c+d x))}{\left (a^2-b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 64, normalized size = 0.85 \[ \frac {(a-b) \log (1-\cos (c+d x))+(a+b) \log (\cos (c+d x)+1)-2 a \log (a \cos (c+d x)+b)}{2 d (a-b) (a+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 65, normalized size = 0.87 \[ -\frac {2 \, a \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a + b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a - b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 101, normalized size = 1.35 \[ -\frac {\frac {2 \, a \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} - b^{2}} - \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 75, normalized size = 1.00 \[ -\frac {a \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right ) \left (a -b \right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}+\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 73, normalized size = 0.97 \[ -\frac {\frac {a \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{2} - b^{2}} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a + b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.71, size = 68, normalized size = 0.91 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (a+b\right )}-\frac {a\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\left (c + d x \right )}}{a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________