Optimal. Leaf size=102 \[ \frac {1}{2} a^2 (2 B+3 C) x+\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4157, 4098,
3873, 2717, 4130, 8} \begin {gather*} \frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (2 B+3 C)+\frac {B \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2717
Rule 3873
Rule 4098
Rule 4130
Rule 4157
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 B+3 C) \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx\\ &=\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 B+3 C) \int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (2 a^2 (2 B+3 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{2} \left (a^2 (2 B+3 C)\right ) \int 1 \, dx\\ &=\frac {1}{2} a^2 (2 B+3 C) x+\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 61, normalized size = 0.60 \begin {gather*} \frac {a^2 (12 B d x+18 C d x+3 (7 B+8 C) \sin (c+d x)+3 (2 B+C) \sin (2 (c+d x))+B \sin (3 (c+d x)))}{12 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.04, size = 116, normalized size = 1.14
method | result | size |
risch | \(a^{2} B x +\frac {3 a^{2} x C}{2}+\frac {7 a^{2} B \sin \left (d x +c \right )}{4 d}+\frac {2 \sin \left (d x +c \right ) a^{2} C}{d}+\frac {a^{2} B \sin \left (3 d x +3 c \right )}{12 d}+\frac {a^{2} B \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 d}\) | \(99\) |
derivativedivides | \(\frac {\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} B \sin \left (d x +c \right )+2 a^{2} C \sin \left (d x +c \right )+a^{2} C \left (d x +c \right )}{d}\) | \(116\) |
default | \(\frac {\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} B \sin \left (d x +c \right )+2 a^{2} C \sin \left (d x +c \right )+a^{2} C \left (d x +c \right )}{d}\) | \(116\) |
norman | \(\frac {\frac {a^{2} \left (2 B +3 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2} \left (2 B +3 C \right ) x}{2}+\frac {13 a^{2} \left (2 B +3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{2} \left (2 B +3 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{2} \left (2 B +3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (2 B +3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (2 B +3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a^{2} \left (2 B +3 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a^{2} \left (2 B +3 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (2 B +3 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (2 B +3 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a^{2} \left (6 B +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{2} \left (10 B +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{2} \left (14 B +33 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 \left (2 B +C \right ) a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(391\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 110, normalized size = 1.08 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 12 \, {\left (d x + c\right )} C a^{2} - 12 \, B a^{2} \sin \left (d x + c\right ) - 24 \, C a^{2} \sin \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.61, size = 70, normalized size = 0.69 \begin {gather*} \frac {3 \, {\left (2 \, B + 3 \, C\right )} a^{2} d x + {\left (2 \, B a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (5 \, B + 6 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.50, size = 142, normalized size = 1.39 \begin {gather*} \frac {3 \, {\left (2 \, B a^{2} + 3 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{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 )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.87, size = 98, normalized size = 0.96 \begin {gather*} B\,a^2\,x+\frac {3\,C\,a^2\,x}{2}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________