Optimal. Leaf size=162 \[ \frac{5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac{a^3 (28 A+20 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(12 A+20 B+15 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{24 d}+a^3 A x+\frac{(4 B+3 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 a d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237257, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4054, 3917, 3914, 3767, 8, 3770} \[ \frac{5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac{a^3 (28 A+20 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(12 A+20 B+15 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{24 d}+a^3 A x+\frac{(4 B+3 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 a d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4054
Rule 3917
Rule 3914
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{\int (a+a \sec (c+d x))^3 (4 a A+a (4 B+3 C) \sec (c+d x)) \, dx}{4 a}\\ &=\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac{\int (a+a \sec (c+d x))^2 \left (12 a^2 A+a^2 (12 A+20 B+15 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac{(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{\int (a+a \sec (c+d x)) \left (24 a^3 A+15 a^3 (4 A+4 B+3 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=a^3 A x+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac{(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{1}{8} \left (5 a^3 (4 A+4 B+3 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{8} \left (a^3 (28 A+20 B+15 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 A x+\frac{a^3 (28 A+20 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac{(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac{\left (5 a^3 (4 A+4 B+3 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d}\\ &=a^3 A x+\frac{a^3 (28 A+20 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac{(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}\\ \end{align*}
Mathematica [B] time = 3.09256, size = 464, normalized size = 2.86 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (\sec (c) (12 A \sin (2 c+d x)+216 A \sin (c+2 d x)-72 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+72 A \sin (3 c+4 d x)+72 A d x \cos (c)+48 A d x \cos (c+2 d x)+48 A d x \cos (3 c+2 d x)+12 A d x \cos (3 c+4 d x)+12 A d x \cos (5 c+4 d x)-216 A \sin (c)+12 A \sin (d x)+36 B \sin (2 c+d x)+280 B \sin (c+2 d x)-72 B \sin (3 c+2 d x)+36 B \sin (2 c+3 d x)+36 B \sin (4 c+3 d x)+88 B \sin (3 c+4 d x)-264 B \sin (c)+36 B \sin (d x)+69 C \sin (2 c+d x)+264 C \sin (c+2 d x)-24 C \sin (3 c+2 d x)+45 C \sin (2 c+3 d x)+45 C \sin (4 c+3 d x)+72 C \sin (3 c+4 d x)-216 C \sin (c)+69 C \sin (d x))-24 (28 A+20 B+15 C) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{768 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.065, size = 262, normalized size = 1.6 \begin{align*}{a}^{3}Ax+{\frac{A{a}^{3}c}{d}}+{\frac{5\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{{a}^{3}C\tan \left ( dx+c \right ) }{d}}+{\frac{7\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{11\,B{a}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{15\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+3\,{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{3\,B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{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}}{3\,d}}+{\frac{{a}^{3}C\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 = 0.960431, size = 475, normalized size = 2.93 \begin{align*} \frac{48 \,{\left (d x + c\right )} A a^{3} + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 48 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, C 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 )} - 12 \, 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 )} - 36 \, C 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 )} + 144 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, B 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 ) + 144 \, B a^{3} \tan \left (d x + c\right ) + 48 \, C 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 = 0.558428, size = 447, normalized size = 2.76 \begin{align*} \frac{48 \, A a^{3} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (28 \, A + 20 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (28 \, A + 20 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (9 \, A + 11 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, A + 12 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, C 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\, dx + \int 3 A \sec{\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec{\left (c + d x \right )}\, dx + \int 3 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \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 = 1.32838, size = 406, normalized size = 2.51 \begin{align*} \frac{24 \,{\left (d x + c\right )} A a^{3} + 3 \,{\left (28 \, A a^{3} + 20 \, B a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (28 \, A a^{3} + 20 \, B a^{3} + 15 \, C 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} + 60 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 45 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 204 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 220 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 165 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 228 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 292 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 219 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 84 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 132 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 147 \, C 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]