Optimal. Leaf size=150 \[ \frac {\left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {\left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (6 a^2+5 b^2\right )+\frac {2 a b \sin ^5(c+d x)}{5 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {\left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {\left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (6 a^2+5 b^2\right )+\frac {2 a b \sin ^5(c+d x)}{5 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2789
Rule 3014
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \cos (c+d x))^2 \, dx &=(2 a b) \int \cos ^5(c+d x) \, dx+\int \cos ^4(c+d x) \left (a^2+b^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac {b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} \left (6 a^2+5 b^2\right ) \int \cos ^4(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {2 a b \sin (c+d x)}{d}+\frac {\left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin ^5(c+d x)}{5 d}+\frac {1}{8} \left (6 a^2+5 b^2\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {2 a b \sin (c+d x)}{d}+\frac {\left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin ^5(c+d x)}{5 d}+\frac {1}{16} \left (6 a^2+5 b^2\right ) \int 1 \, dx\\ &=\frac {1}{16} \left (6 a^2+5 b^2\right ) x+\frac {2 a b \sin (c+d x)}{d}+\frac {\left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 123, normalized size = 0.82 \[ \frac {5 \left (\left (48 a^2+45 b^2\right ) \sin (2 (c+d x))+\left (6 a^2+9 b^2\right ) \sin (4 (c+d x))+72 a^2 c+72 a^2 d x+b^2 \sin (6 (c+d x))+60 b^2 c+60 b^2 d x\right )+384 a b \sin ^5(c+d x)-1280 a b \sin ^3(c+d x)+1920 a b \sin (c+d x)}{960 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.98, size = 110, normalized size = 0.73 \[ \frac {15 \, {\left (6 \, a^{2} + 5 \, b^{2}\right )} d x + {\left (40 \, b^{2} \cos \left (d x + c\right )^{5} + 96 \, a b \cos \left (d x + c\right )^{4} + 128 \, a b \cos \left (d x + c\right )^{2} + 10 \, {\left (6 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 256 \, a b + 15 \, {\left (6 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.55, size = 127, normalized size = 0.85 \[ \frac {1}{16} \, {\left (6 \, a^{2} + 5 \, b^{2}\right )} x + \frac {b^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a b \sin \left (5 \, d x + 5 \, c\right )}{40 \, d} + \frac {5 \, a b \sin \left (3 \, d x + 3 \, c\right )}{24 \, d} + \frac {5 \, a b \sin \left (d x + c\right )}{4 \, d} + \frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, a^{2} + 15 \, b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 120, normalized size = 0.80 \[ \frac {b^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 120, normalized size = 0.80 \[ \frac {30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 128 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a b - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 143, normalized size = 0.95 \[ \frac {3\,a^2\,x}{8}+\frac {5\,b^2\,x}{16}+\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {15\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {b^2\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,a\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{24\,d}+\frac {a\,b\,\sin \left (5\,c+5\,d\,x\right )}{40\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.02, size = 343, normalized size = 2.29 \[ \begin {cases} \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {16 a b \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {8 a b \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{2} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________