Optimal. Leaf size=72 \[ \frac {d \sin (a+b x) \cos ^3(a+b x)}{16 b^2}+\frac {3 d \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {3 d x}{32 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4405, 2635, 8} \[ \frac {d \sin (a+b x) \cos ^3(a+b x)}{16 b^2}+\frac {3 d \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {3 d x}{32 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 4405
Rubi steps
\begin {align*} \int (c+d x) \cos ^3(a+b x) \sin (a+b x) \, dx &=-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {d \int \cos ^4(a+b x) \, dx}{4 b}\\ &=-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {d \cos ^3(a+b x) \sin (a+b x)}{16 b^2}+\frac {(3 d) \int \cos ^2(a+b x) \, dx}{16 b}\\ &=-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {3 d \cos (a+b x) \sin (a+b x)}{32 b^2}+\frac {d \cos ^3(a+b x) \sin (a+b x)}{16 b^2}+\frac {(3 d) \int 1 \, dx}{32 b}\\ &=\frac {3 d x}{32 b}-\frac {(c+d x) \cos ^4(a+b x)}{4 b}+\frac {3 d \cos (a+b x) \sin (a+b x)}{32 b^2}+\frac {d \cos ^3(a+b x) \sin (a+b x)}{16 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 75, normalized size = 1.04 \[ \frac {d (\sin (2 (a+b x))-2 b x \cos (2 (a+b x)))}{16 b^2}+\frac {d (\sin (4 (a+b x))-4 b x \cos (4 (a+b x)))}{128 b^2}-\frac {c \cos ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 58, normalized size = 0.81 \[ -\frac {8 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{4} - 3 \, b d x - {\left (2 \, d \cos \left (b x + a\right )^{3} + 3 \, d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{32 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 75, normalized size = 1.04 \[ -\frac {{\left (b d x + b c\right )} \cos \left (4 \, b x + 4 \, a\right )}{32 \, b^{2}} - \frac {{\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{2}} + \frac {d \sin \left (4 \, b x + 4 \, a\right )}{128 \, b^{2}} + \frac {d \sin \left (2 \, b x + 2 \, a\right )}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 85, normalized size = 1.18 \[ \frac {\frac {d \left (-\frac {\left (b x +a \right ) \left (\cos ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b}+\frac {d a \left (\cos ^{4}\left (b x +a \right )\right )}{4 b}-\frac {c \left (\cos ^{4}\left (b x +a \right )\right )}{4}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 92, normalized size = 1.28 \[ -\frac {32 \, c \cos \left (b x + a\right )^{4} - \frac {32 \, a d \cos \left (b x + a\right )^{4}}{b} + \frac {{\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} d}{b}}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.31, size = 94, normalized size = 1.31 \[ \frac {4\,d\,\sin \left (2\,a+2\,b\,x\right )+\frac {d\,\sin \left (4\,a+4\,b\,x\right )}{2}+4\,b\,c\,{\sin \left (2\,a+2\,b\,x\right )}^2+16\,b\,c\,{\sin \left (a+b\,x\right )}^2+8\,b\,d\,x\,\left (2\,{\sin \left (a+b\,x\right )}^2-1\right )+2\,b\,d\,x\,\left (2\,{\sin \left (2\,a+2\,b\,x\right )}^2-1\right )}{64\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.94, size = 138, normalized size = 1.92 \[ \begin {cases} - \frac {c \cos ^{4}{\left (a + b x \right )}}{4 b} + \frac {3 d x \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {5 d x \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 d \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{32 b^{2}} + \frac {5 d \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin {\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________