Optimal. Leaf size=102 \[ \frac{(2 A+3 B) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(2 A+3 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.114332, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {4000, 3796, 3794} \[ \frac{(2 A+3 B) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(2 A+3 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4000
Rule 3796
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(2 A+3 B) \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{5 a}\\ &=\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(2 A+3 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(2 A+3 B) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2}\\ &=\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(2 A+3 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(2 A+3 B) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.324102, size = 135, normalized size = 1.32 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-15 (2 A+B) \sin \left (c+\frac{d x}{2}\right )+5 (8 A+3 B) \sin \left (\frac{d x}{2}\right )+20 A \sin \left (c+\frac{3 d x}{2}\right )-15 A \sin \left (2 c+\frac{3 d x}{2}\right )+7 A \sin \left (2 c+\frac{5 d x}{2}\right )+15 B \sin \left (c+\frac{3 d x}{2}\right )+3 B \sin \left (2 c+\frac{5 d x}{2}\right )\right )}{30 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 64, normalized size = 0.6 \begin{align*}{\frac{1}{4\,d{a}^{3}} \left ({\frac{A-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{2\,A}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+A\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +B\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01165, size = 155, normalized size = 1.52 \begin{align*} \frac{\frac{A{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{3 \, B{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.444781, size = 227, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (7 \, A + 3 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (2 \, A + 3 \, B\right )} \cos \left (d x + c\right ) + 2 \, A + 3 \, B\right )} \sin \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\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*} \frac{\int \frac{A \sec{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32072, size = 101, normalized size = 0.99 \begin{align*} \frac{3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 10 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{60 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]