Optimal. Leaf size=153 \[ -\frac{a^2 \cos ^7(c+d x)}{28 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{36 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a^2 x}{64}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.153673, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac{a^2 \cos ^7(c+d x)}{28 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{36 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a^2 x}{64}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2860
Rule 2678
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx &=-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac{2}{9} \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac{1}{4} a \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{a^2 \cos ^7(c+d x)}{28 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac{1}{4} a^2 \int \cos ^6(c+d x) \, dx\\ &=-\frac{a^2 \cos ^7(c+d x)}{28 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac{1}{24} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{a^2 \cos ^7(c+d x)}{28 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac{1}{32} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a^2 \cos ^7(c+d x)}{28 d}+\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac{1}{64} \left (5 a^2\right ) \int 1 \, dx\\ &=\frac{5 a^2 x}{64}-\frac{a^2 \cos ^7(c+d x)}{28 d}+\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}\\ \end{align*}
Mathematica [A] time = 0.645294, size = 106, normalized size = 0.69 \[ \frac{a^2 (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-3276 \cos (c+d x)-1848 \cos (3 (c+d x))-504 \cos (5 (c+d x))-18 \cos (7 (c+d x))+14 \cos (9 (c+d x))+2520 c+2520 d x)}{32256 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 116, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +2\,{a}^{2} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03087, size = 126, normalized size = 0.82 \begin{align*} -\frac{4608 \, a^{2} \cos \left (d x + c\right )^{7} - 512 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 21 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{32256 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.21029, size = 251, normalized size = 1.64 \begin{align*} \frac{448 \, a^{2} \cos \left (d x + c\right )^{9} - 1152 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, a^{2} d x - 21 \,{\left (48 \, a^{2} \cos \left (d x + c\right )^{7} - 8 \, a^{2} \cos \left (d x + c\right )^{5} - 10 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4032 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 22.3364, size = 282, normalized size = 1.84 \begin{align*} \begin{cases} \frac{5 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac{5 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac{5 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac{5 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac{5 a^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{64 d} + \frac{55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac{73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac{a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{5 a^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac{2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac{a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} \sin{\left (c \right )} \cos ^{6}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22829, size = 212, normalized size = 1.39 \begin{align*} \frac{5}{64} \, a^{2} x + \frac{a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac{a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{a^{2} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac{11 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{13 \, a^{2} \cos \left (d x + c\right )}{128 \, d} - \frac{a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac{a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]