Optimal. Leaf size=52 \[ \frac {(a C+b B) \sin (c+d x)}{d}+\frac {1}{2} x (a B+2 b C)+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {4072, 3996, 3787, 2637, 8} \[ \frac {(a C+b B) \sin (c+d x)}{d}+\frac {1}{2} x (a B+2 b C)+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2637
Rule 3787
Rule 3996
Rule 4072
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) (-2 (b B+a C)-(a B+2 b C) \sec (c+d x)) \, dx\\ &=\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}-(-b B-a C) \int \cos (c+d x) \, dx-\frac {1}{2} (-a B-2 b C) \int 1 \, dx\\ &=\frac {1}{2} (a B+2 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 51, normalized size = 0.98 \[ \frac {4 (a C+b B) \sin (c+d x)+a B \sin (2 (c+d x))+2 a B c+2 a B d x+4 b C d x}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 42, normalized size = 0.81 \[ \frac {{\left (B a + 2 \, C b\right )} d x + {\left (B a \cos \left (d x + c\right ) + 2 \, C a + 2 \, B b\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 121, normalized size = 2.33 \[ \frac {{\left (B a + 2 \, C b\right )} {\left (d x + c\right )} - \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.90, size = 57, normalized size = 1.10 \[ \frac {a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B b \sin \left (d x +c \right )+a C \sin \left (d x +c \right )+C b \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 55, normalized size = 1.06 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, {\left (d x + c\right )} C b + 4 \, C a \sin \left (d x + c\right ) + 4 \, B b \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.65, size = 50, normalized size = 0.96 \[ \frac {B\,a\,x}{2}+C\,b\,x+\frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________