Optimal. Leaf size=156 \[ \frac {\left (2 a^2 C+4 a b B+3 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {\left (3 a^2 B+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (3 a^2 B+8 a b C+4 b^2 B\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a^2 B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {a (a C+2 b B) \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3029, 2988, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac {\left (2 a^2 C+4 a b B+3 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {\left (3 a^2 B+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (3 a^2 B+8 a b C+4 b^2 B\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a^2 B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {a (a C+2 b B) \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2748
Rule 2988
Rule 3021
Rule 3029
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\frac {a^2 B \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{4} \int \left (-4 a (2 b B+a C)-\left (3 a^2 B+4 b^2 B+8 a b C\right ) \cos (c+d x)-4 b^2 C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {a (2 b B+a C) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a^2 B \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{12} \int \left (-3 \left (3 a^2 B+4 b^2 B+8 a b C\right )-4 \left (4 a b B+2 a^2 C+3 b^2 C\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a (2 b B+a C) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a^2 B \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{4} \left (-3 a^2 B-4 b^2 B-8 a b C\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{3} \left (-4 a b B-2 a^2 C-3 b^2 C\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {\left (3 a^2 B+4 b^2 B+8 a b C\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a (2 b B+a C) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a^2 B \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-3 a^2 B-4 b^2 B-8 a b C\right ) \int \sec (c+d x) \, dx-\frac {\left (4 a b B+2 a^2 C+3 b^2 C\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {\left (3 a^2 B+4 b^2 B+8 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (4 a b B+2 a^2 C+3 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {\left (3 a^2 B+4 b^2 B+8 a b C\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a (2 b B+a C) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a^2 B \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.74, size = 120, normalized size = 0.77 \[ \frac {3 \left (3 a^2 B+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 \left (3 a^2 B+8 a b C+4 b^2 B\right ) \sec (c+d x)+24 \left (a^2 C+2 a b B+b^2 C\right )+6 a^2 B \sec ^3(c+d x)+8 a (a C+2 b B) \tan ^2(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 180, normalized size = 1.15 \[ \frac {3 \, {\left (3 \, B a^{2} + 8 \, C a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, B a^{2} + 8 \, C a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (2 \, C a^{2} + 4 \, B a b + 3 \, C b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, B a^{2} + 3 \, {\left (3 \, B a^{2} + 8 \, C a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.25, size = 478, normalized size = 3.06 \[ \frac {3 \, {\left (3 \, B a^{2} + 8 \, C a b + 4 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, B a^{2} + 8 \, C a b + 4 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 80 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C b^{2} \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} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.39, size = 241, normalized size = 1.54 \[ \frac {2 a^{2} C \tan \left (d x +c \right )}{3 d}+\frac {a^{2} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a^{2} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {C a b \tan \left (d x +c \right ) \sec \left (d x +c \right )}{d}+\frac {C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 B a b \tan \left (d x +c \right )}{3 d}+\frac {2 B a b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} C \tan \left (d x +c \right )}{d}+\frac {b^{2} B \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 228, normalized size = 1.46 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} + 32 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b - 3 \, B a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, C a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b^{2} \tan \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.28, size = 314, normalized size = 2.01 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,B\,a^2}{8}+C\,a\,b+\frac {B\,b^2}{2}\right )}{\frac {3\,B\,a^2}{2}+4\,C\,a\,b+2\,B\,b^2}\right )\,\left (\frac {3\,B\,a^2}{4}+2\,C\,a\,b+B\,b^2\right )}{d}+\frac {\left (\frac {5\,B\,a^2}{4}+B\,b^2-2\,C\,a^2-2\,C\,b^2-4\,B\,a\,b+2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,B\,a^2}{4}-B\,b^2+\frac {10\,C\,a^2}{3}+6\,C\,b^2+\frac {20\,B\,a\,b}{3}-2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,B\,a^2}{4}-B\,b^2-\frac {10\,C\,a^2}{3}-6\,C\,b^2-\frac {20\,B\,a\,b}{3}-2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,B\,a^2}{4}+B\,b^2+2\,C\,a^2+2\,C\,b^2+4\,B\,a\,b+2\,C\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________