Optimal. Leaf size=158 \[ -\frac{a \left (a^2+b^2\right ) \tan (c+d x)}{b^4 d}+\frac{\left (a^2+b^2\right ) \sec ^2(c+d x)}{2 b^3 d}-\frac{\left (a^2+b^2\right )^2 \log (\cos (c+d x))}{b^5 d}+\frac{\left (a^2+b^2\right )^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b^5 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^4(c+d x)}{4 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.222416, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3104, 3767, 8, 3102, 3475, 3133} \[ -\frac{a \left (a^2+b^2\right ) \tan (c+d x)}{b^4 d}+\frac{\left (a^2+b^2\right ) \sec ^2(c+d x)}{2 b^3 d}-\frac{\left (a^2+b^2\right )^2 \log (\cos (c+d x))}{b^5 d}+\frac{\left (a^2+b^2\right )^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b^5 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^4(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3104
Rule 3767
Rule 8
Rule 3102
Rule 3475
Rule 3133
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx &=\frac{\sec ^4(c+d x)}{4 b d}-\frac{a \int \sec ^4(c+d x) \, dx}{b^2}+\frac{\left (a^2+b^2\right ) \int \frac{\sec ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^2}\\ &=\frac{\left (a^2+b^2\right ) \sec ^2(c+d x)}{2 b^3 d}+\frac{\sec ^4(c+d x)}{4 b d}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \sec ^2(c+d x) \, dx}{b^4}+\frac{\left (a^2+b^2\right )^2 \int \frac{\sec (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4}+\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{b^2 d}\\ &=\frac{\left (a^2+b^2\right ) \sec ^2(c+d x)}{2 b^3 d}+\frac{\sec ^4(c+d x)}{4 b d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}+\frac{\left (a^2+b^2\right )^2 \int \frac{b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^5}+\frac{\left (a^2+b^2\right )^2 \int \tan (c+d x) \, dx}{b^5}+\frac{\left (a \left (a^2+b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{b^4 d}\\ &=-\frac{\left (a^2+b^2\right )^2 \log (\cos (c+d x))}{b^5 d}+\frac{\left (a^2+b^2\right )^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b^5 d}+\frac{\left (a^2+b^2\right ) \sec ^2(c+d x)}{2 b^3 d}+\frac{\sec ^4(c+d x)}{4 b d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{a \left (a^2+b^2\right ) \tan (c+d x)}{b^4 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}\\ \end{align*}
Mathematica [A] time = 1.20875, size = 99, normalized size = 0.63 \[ \frac{6 b^2 \left (a^2+b^2\right ) \tan ^2(c+d x)-12 a b \left (a^2+2 b^2\right ) \tan (c+d x)+12 \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))-4 a b^3 \tan ^3(c+d x)+3 b^4 \sec ^4(c+d x)}{12 b^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.158, size = 162, normalized size = 1. \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,db}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d{b}^{3}}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{db}}-{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d{b}^{4}}}-2\,{\frac{a\tan \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){a}^{4}}{d{b}^{5}}}+2\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{db}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.09426, size = 624, normalized size = 3.95 \begin{align*} -\frac{\frac{2 \,{\left (\frac{3 \,{\left (a^{3} + 2 \, a b^{2}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{{\left (9 \, a^{3} + 14 \, a b^{2}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{6 \,{\left (a^{2} b + b^{3}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{{\left (9 \, a^{3} + 14 \, a b^{2}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{3 \,{\left (a^{3} + 2 \, a b^{2}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{b^{4} - \frac{4 \, b^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, b^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, b^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{b^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \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 )}{b^{5}} + \frac{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{5}} + \frac{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{5}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.568298, size = 439, normalized size = 2.78 \begin{align*} \frac{6 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} \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 ) - 6 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\cos \left (d x + c\right )^{2}\right ) + 3 \, b^{4} + 6 \,{\left (a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \,{\left (a b^{3} \cos \left (d x + c\right ) +{\left (3 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} d \cos \left (d x + c\right )^{4}} \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 ^{5}{\left (c + d x \right )}}{a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19942, size = 162, normalized size = 1.03 \begin{align*} \frac{\frac{3 \, b^{3} \tan \left (d x + c\right )^{4} - 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 12 \, b^{3} \tan \left (d x + c\right )^{2} - 12 \, a^{3} \tan \left (d x + c\right ) - 24 \, a b^{2} \tan \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]