Optimal. Leaf size=174 \[ \frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^2 x}{16}-\frac{a b \cos ^6(c+d x)}{3 d}-\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^2 x}{16} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.170358, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3090, 2635, 8, 2565, 30, 2568} \[ \frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^2 x}{16}-\frac{a b \cos ^6(c+d x)}{3 d}-\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^2 x}{16} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3090
Rule 2635
Rule 8
Rule 2565
Rule 30
Rule 2568
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^6(c+d x)+2 a b \cos ^5(c+d x) \sin (c+d x)+b^2 \cos ^4(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^6(c+d x) \, dx+(2 a b) \int \cos ^5(c+d x) \sin (c+d x) \, dx+b^2 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx\\ &=\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac{b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{6} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{6} b^2 \int \cos ^4(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int x^5 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac{b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{8} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{8} b^2 \int \cos ^2(c+d x) \, dx\\ &=-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac{b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{16} \left (5 a^2\right ) \int 1 \, dx+\frac{1}{16} b^2 \int 1 \, dx\\ &=\frac{5 a^2 x}{16}+\frac{b^2 x}{16}-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac{b^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.24063, size = 147, normalized size = 0.84 \[ \frac{\left (5 a^2+b^2\right ) (c+d x)}{16 d}+\frac{\left (15 a^2+b^2\right ) \sin (2 (c+d x))}{64 d}+\frac{\left (3 a^2-b^2\right ) \sin (4 (c+d x))}{64 d}+\frac{\left (a^2-b^2\right ) \sin (6 (c+d x))}{192 d}-\frac{5 a b \cos (2 (c+d x))}{32 d}-\frac{a b \cos (4 (c+d x))}{16 d}-\frac{a b \cos (6 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 118, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{\sin \left ( dx+c \right ) }{24} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{dx}{16}}+{\frac{c}{16}} \right ) -{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3}}+{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.22801, size = 138, normalized size = 0.79 \begin{align*} -\frac{64 \, a b \cos \left (d x + c\right )^{6} +{\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 )} a^{2} -{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.503055, size = 223, normalized size = 1.28 \begin{align*} -\frac{16 \, a b \cos \left (d x + c\right )^{6} - 3 \,{\left (5 \, a^{2} + b^{2}\right )} d x -{\left (8 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (5 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (5 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.89803, size = 384, normalized size = 2.21 \begin{align*} \begin{cases} \frac{5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac{11 a^{2} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac{a b \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{a b \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{a b \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{3 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{3 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{b^{2} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac{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 \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{2} \cos ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15153, size = 178, normalized size = 1.02 \begin{align*} \frac{1}{16} \,{\left (5 \, a^{2} + b^{2}\right )} x - \frac{a b \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac{a b \cos \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac{5 \, a b \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac{{\left (a^{2} - b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (3 \, a^{2} - b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (15 \, a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]