Optimal. Leaf size=196 \[ \frac{5 a^4 (4 A+8 B+7 C) \tan (c+d x)}{8 d}+\frac{a^4 (52 A+48 B+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{(12 A-4 B-7 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 d}-\frac{(12 A-32 B-35 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+a^4 x (4 A+B)-\frac{a (4 A-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^4}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.382038, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4086, 3917, 3914, 3767, 8, 3770} \[ \frac{5 a^4 (4 A+8 B+7 C) \tan (c+d x)}{8 d}+\frac{a^4 (52 A+48 B+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{(12 A-4 B-7 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 d}-\frac{(12 A-32 B-35 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+a^4 x (4 A+B)-\frac{a (4 A-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^4}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4086
Rule 3917
Rule 3914
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{\int (a+a \sec (c+d x))^4 (a (4 A+B)-a (4 A-C) \sec (c+d x)) \, dx}{a}\\ &=\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{\int (a+a \sec (c+d x))^3 \left (4 a^2 (4 A+B)-a^2 (12 A-4 B-7 C) \sec (c+d x)\right ) \, dx}{4 a}\\ &=\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{(12 A-4 B-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}+\frac{\int (a+a \sec (c+d x))^2 \left (12 a^3 (4 A+B)-a^3 (12 A-32 B-35 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{(12 A-4 B-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac{(12 A-32 B-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{\int (a+a \sec (c+d x)) \left (24 a^4 (4 A+B)+15 a^4 (4 A+8 B+7 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=a^4 (4 A+B) x+\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{(12 A-4 B-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac{(12 A-32 B-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{1}{8} \left (5 a^4 (4 A+8 B+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{8} \left (a^4 (52 A+48 B+35 C)\right ) \int \sec (c+d x) \, dx\\ &=a^4 (4 A+B) x+\frac{a^4 (52 A+48 B+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{(12 A-4 B-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac{(12 A-32 B-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac{\left (5 a^4 (4 A+8 B+7 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d}\\ &=a^4 (4 A+B) x+\frac{a^4 (52 A+48 B+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+a \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{5 a^4 (4 A+8 B+7 C) \tan (c+d x)}{8 d}-\frac{a (4 A-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{(12 A-4 B-7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}-\frac{(12 A-32 B-35 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}\\ \end{align*}
Mathematica [B] time = 4.65642, size = 530, normalized size = 2.7 \[ \frac{a^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (\sec (c) (72 d x (4 A+B) \cos (c)+48 d x (4 A+B) \cos (c+2 d x)+24 A \sin (2 c+d x)+288 A \sin (c+2 d x)-96 A \sin (3 c+2 d x)+30 A \sin (2 c+3 d x)+30 A \sin (4 c+3 d x)+96 A \sin (3 c+4 d x)+6 A \sin (4 c+5 d x)+6 A \sin (6 c+5 d x)+192 A d x \cos (3 c+2 d x)+48 A d x \cos (3 c+4 d x)+48 A d x \cos (5 c+4 d x)-288 A \sin (c)+24 A \sin (d x)+48 B \sin (2 c+d x)+496 B \sin (c+2 d x)-144 B \sin (3 c+2 d x)+48 B \sin (2 c+3 d x)+48 B \sin (4 c+3 d x)+160 B \sin (3 c+4 d x)+48 B d x \cos (3 c+2 d x)+12 B d x \cos (3 c+4 d x)+12 B d x \cos (5 c+4 d x)-480 B \sin (c)+48 B \sin (d x)+105 C \sin (2 c+d x)+544 C \sin (c+2 d x)-96 C \sin (3 c+2 d x)+81 C \sin (2 c+3 d x)+81 C \sin (4 c+3 d x)+160 C \sin (3 c+4 d x)-480 C \sin (c)+105 C \sin (d x))-24 (52 A+48 B+35 C) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1536 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.117, size = 294, normalized size = 1.5 \begin{align*}{\frac{A{a}^{4}\sin \left ( dx+c \right ) }{d}}+B{a}^{4}x+{\frac{B{a}^{4}c}{d}}+{\frac{35\,{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+4\,{a}^{4}Ax+4\,{\frac{A{a}^{4}c}{d}}+6\,{\frac{B{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{20\,{a}^{4}C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{13\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{20\,B{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{27\,{a}^{4}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+4\,{\frac{A{a}^{4}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{B{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{4\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{B{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4}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 = 0.978593, size = 562, normalized size = 2.87 \begin{align*} \frac{192 \,{\left (d x + c\right )} A a^{4} + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 48 \,{\left (d x + c\right )} B a^{4} + 64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 3 \, C a^{4}{\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^{4}{\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 \, B a^{4}{\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 )} - 72 \, C a^{4}{\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^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, A a^{4} \tan \left (d x + c\right ) + 288 \, B a^{4} \tan \left (d x + c\right ) + 192 \, C a^{4} \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 = 0.565522, size = 493, normalized size = 2.52 \begin{align*} \frac{48 \,{\left (4 \, A + B\right )} a^{4} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (52 \, A + 48 \, B + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (52 \, A + 48 \, B + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \,{\left (3 \, A + 5 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, A + 16 \, B + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \,{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 6 \, C a^{4}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32555, size = 458, normalized size = 2.34 \begin{align*} \frac{\frac{48 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 24 \,{\left (4 \, A a^{4} + B a^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (52 \, A a^{4} + 48 \, B a^{4} + 35 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (52 \, A a^{4} + 48 \, B a^{4} + 35 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (84 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 276 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 424 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 385 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 108 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 216 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 279 \, C a^{4} \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]