Optimal. Leaf size=108 \[ -\frac{a^3 \log (a \cos (c+d x)+b)}{b^2 d \left (a^2-b^2\right )}+\frac{a \log (\cos (c+d x))}{b^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\sec (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.284128, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4397, 2837, 12, 894} \[ -\frac{a^3 \log (a \cos (c+d x)+b)}{b^2 d \left (a^2-b^2\right )}+\frac{a \log (\cos (c+d x))}{b^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\sec (c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx &=\int \frac{\csc (c+d x) \sec ^2(c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{a^2}{x^2 (b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x^2 (b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{2 a^3 (a+b) (a-x)}+\frac{1}{a^2 b x^2}-\frac{1}{a^2 b^2 x}-\frac{1}{2 a^3 (a-b) (a+x)}-\frac{1}{b^2 (-a+b) (a+b) (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac{\log (1-\cos (c+d x))}{2 (a+b) d}+\frac{a \log (\cos (c+d x))}{b^2 d}+\frac{\log (1+\cos (c+d x))}{2 (a-b) d}-\frac{a^3 \log (b+a \cos (c+d x))}{b^2 \left (a^2-b^2\right ) d}+\frac{\sec (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.292099, size = 92, normalized size = 0.85 \[ \frac{\frac{a^3 \log (a \cos (c+d x)+b)}{b^4-a^2 b^2}+\frac{a \log (\cos (c+d x))}{b^2}+\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{a+b}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a-b}+\frac{\sec (c+d x)}{b}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.132, size = 110, normalized size = 1. \begin{align*}{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{d \left ( 2\,a-2\,b \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{d \left ( 2\,a+2\,b \right ) }}+{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{{b}^{2}d}}+{\frac{1}{db\cos \left ( dx+c \right ) }}-{\frac{{a}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) \left ( a-b \right ){b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0753, size = 213, normalized size = 1.97 \begin{align*} -\frac{\frac{a^{3} \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} - b^{4}} - \frac{a \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{2}} - \frac{a \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{2}} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a + b} - \frac{2}{b - \frac{b \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.911594, size = 360, normalized size = 3.33 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right ) \log \left (a \cos \left (d x + c\right ) + b\right ) - 2 \, a^{2} b + 2 \, b^{3} - 2 \,{\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) -{\left (a b^{2} + b^{3}\right )} \cos \left (d x + c\right ) \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a b^{2} - b^{3}\right )} \cos \left (d x + c\right ) \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left (a^{2} b^{2} - b^{4}\right )} d \cos \left (d x + c\right )} \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{\sec ^{3}{\left (c + d x \right )}}{a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26384, size = 257, normalized size = 2.38 \begin{align*} -\frac{\frac{2 \, a^{3} \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} - b^{4}} - \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} - \frac{2 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{b^{2}} + \frac{2 \,{\left (a - 2 \, b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{b^{2}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]