Optimal. Leaf size=125 \[ \frac{a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac{5 a^3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.142558, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {4001, 3791, 3770, 3767, 8, 3768} \[ \frac{a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac{5 a^3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4001
Rule 3791
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{4} (4 A+3 B) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx\\ &=\frac{B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{4} (4 A+3 B) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx\\ &=\frac{B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{4} \left (a^3 (4 A+3 B)\right ) \int \sec (c+d x) \, dx+\frac{1}{4} \left (a^3 (4 A+3 B)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{3 a^3 (4 A+3 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{8} \left (3 a^3 (4 A+3 B)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (4 A+3 B)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{4 d}-\frac{\left (3 a^3 (4 A+3 B)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{4 d}\\ &=\frac{5 a^3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac{3 a^3 (4 A+3 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d}\\ \end{align*}
Mathematica [B] time = 1.29325, size = 273, normalized size = 2.18 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (120 (4 A+3 B) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-24 (11 A+9 B) \sin (c)+(36 A+69 B) \sin (d x)+36 A \sin (2 c+d x)+280 A \sin (c+2 d x)-72 A \sin (3 c+2 d x)+36 A \sin (2 c+3 d x)+36 A \sin (4 c+3 d x)+88 A \sin (3 c+4 d x)+69 B \sin (2 c+d x)+264 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+45 B \sin (2 c+3 d x)+45 B \sin (4 c+3 d x)+72 B \sin (3 c+4 d x))\right )}{1536 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 188, normalized size = 1.5 \begin{align*}{\frac{5\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{B{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{11\,A{a}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{15\,B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.00015, size = 354, normalized size = 2.83 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 48 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 3 \, B a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 144 \, A a^{3} \tan \left (d x + c\right ) + 48 \, B a^{3} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.491931, size = 366, normalized size = 2.93 \begin{align*} \frac{15 \,{\left (4 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (4 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (11 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 9 \,{\left (4 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 6 \, B a^{3}\right )} \sin \left (d x + c\right )}{48 \, 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*} a^{3} \left (\int A \sec{\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35507, size = 286, normalized size = 2.29 \begin{align*} \frac{15 \,{\left (4 \, A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (4 \, A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (60 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 45 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 220 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 165 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 292 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 219 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 132 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 147 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]