Optimal. Leaf size=210 \[ \frac{a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac{a^3 (26 A+23 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (22 A+21 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.399342, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4018, 3997, 3787, 3768, 3770, 3767} \[ \frac{a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac{a^3 (26 A+23 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (22 A+21 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4018
Rule 3997
Rule 3787
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{1}{6} \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (3 a (2 A+B)+2 a (3 A+4 B) \sec (c+d x)) \, dx\\ &=\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{30} \int \sec ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (16 A+13 B)+3 a^2 (22 A+21 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{120} \int \sec ^3(c+d x) \left (15 a^3 (26 A+23 B)+24 a^3 (19 A+17 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{5} \left (a^3 (19 A+17 B)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{8} \left (a^3 (26 A+23 B)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^3 (26 A+23 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{16} \left (a^3 (26 A+23 B)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (19 A+17 B)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{a^3 (26 A+23 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac{a^3 (26 A+23 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 2.041, size = 346, normalized size = 1.65 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (480 (26 A+23 B) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-320 (19 A+17 B) \sin (c)+750 (2 A+3 B) \sin (d x)+1500 A \sin (2 c+d x)+7680 A \sin (c+2 d x)-1440 A \sin (3 c+2 d x)+1890 A \sin (2 c+3 d x)+1890 A \sin (4 c+3 d x)+3648 A \sin (3 c+4 d x)+390 A \sin (4 c+5 d x)+390 A \sin (6 c+5 d x)+608 A \sin (5 c+6 d x)+2250 B \sin (2 c+d x)+7680 B \sin (c+2 d x)-480 B \sin (3 c+2 d x)+1955 B \sin (2 c+3 d x)+1955 B \sin (4 c+3 d x)+3264 B \sin (3 c+4 d x)+345 B \sin (4 c+5 d x)+345 B \sin (6 c+5 d x)+544 B \sin (5 c+6 d x))\right )}{61440 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.057, size = 281, normalized size = 1.3 \begin{align*}{\frac{13\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{13\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{34\,B{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{17\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{38\,A{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{19\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{23\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{23\,B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{23\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{3\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02866, size = 547, normalized size = 2.6 \begin{align*} \frac{32 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 96 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 160 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 5 \, B a^{3}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A 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 )} - 90 \, 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 )} - 120 \, 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 )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.50713, size = 485, normalized size = 2.31 \begin{align*} \frac{15 \,{\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (32 \,{\left (19 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \,{\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \,{\left (19 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \,{\left (18 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 48 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 40 \, B a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.39501, size = 378, normalized size = 1.8 \begin{align*} \frac{15 \,{\left (26 \, A a^{3} + 23 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (26 \, A a^{3} + 23 \, 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 (390 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 345 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 2210 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1955 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 5148 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 4554 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5988 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5814 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4190 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3165 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1530 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1575 \, 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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]