Optimal. Leaf size=81 \[ -\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a x}{8}+\frac {b \cos ^5(c+d x)}{5 d}-\frac {b \cos ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 14} \[ -\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a x}{8}+\frac {b \cos ^5(c+d x)}{5 d}-\frac {b \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 14
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx+b \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} a \int \cos ^2(c+d x) \, dx-\frac {b \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} a \int 1 \, dx-\frac {b \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a x}{8}-\frac {b \cos ^3(c+d x)}{3 d}+\frac {b \cos ^5(c+d x)}{5 d}+\frac {a \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 59, normalized size = 0.73 \[ \frac {-15 a \sin (4 (c+d x))+60 a c+60 a d x-60 b \cos (c+d x)-10 b \cos (3 (c+d x))+6 b \cos (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 62, normalized size = 0.77 \[ \frac {24 \, b \cos \left (d x + c\right )^{5} - 40 \, b \cos \left (d x + c\right )^{3} + 15 \, a d x - 15 \, {\left (2 \, a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 62, normalized size = 0.77 \[ \frac {1}{8} \, a x + \frac {b \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {b \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {b \cos \left (d x + c\right )}{8 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 77, normalized size = 0.95 \[ \frac {a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 52, normalized size = 0.64 \[ \frac {15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a + 32 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} b}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.89, size = 125, normalized size = 1.54 \[ \frac {a\,x}{8}-\frac {-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {4\,b}{15}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.81, size = 144, normalized size = 1.78 \[ \begin {cases} \frac {a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 b \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin ^{2}{\relax (c )} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________