Optimal. Leaf size=101 \[ \frac {\left (4 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (4 a^2+3 b^2\right )-\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2789, 2633, 3014, 2635, 8} \[ \frac {\left (4 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (4 a^2+3 b^2\right )-\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2789
Rule 3014
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \, dx &=(2 a b) \int \cos ^3(c+d x) \, dx+\int \cos ^2(c+d x) \left (a^2+b^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac {b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \left (4 a^2+3 b^2\right ) \int \cos ^2(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {2 a b \sin (c+d x)}{d}+\frac {\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {1}{8} \left (4 a^2+3 b^2\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (4 a^2+3 b^2\right ) x+\frac {2 a b \sin (c+d x)}{d}+\frac {\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a b \sin ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 86, normalized size = 0.85 \[ \frac {24 \left (a^2+b^2\right ) \sin (2 (c+d x))+48 a^2 c+48 a^2 d x-64 a b \sin ^3(c+d x)+192 a b \sin (c+d x)+3 b^2 \sin (4 (c+d x))+36 b^2 c+36 b^2 d x}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 77, normalized size = 0.76 \[ \frac {3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} d x + {\left (6 \, b^{2} \cos \left (d x + c\right )^{3} + 16 \, a b \cos \left (d x + c\right )^{2} + 32 \, a b + 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 82, normalized size = 0.81 \[ \frac {1}{8} \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} x + \frac {b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a b \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac {3 \, a b \sin \left (d x + c\right )}{2 \, d} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 89, normalized size = 0.88 \[ \frac {b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.83, size = 82, normalized size = 0.81 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a b + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.60, size = 93, normalized size = 0.92 \[ \frac {a^2\,x}{2}+\frac {3\,b^2\,x}{8}+\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.06, size = 211, normalized size = 2.09 \[ \begin {cases} \frac {a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{2} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________