Optimal. Leaf size=124 \[ -\frac {64 a^3 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac {16 a^2 \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}-\frac {2 a \cos ^5(c+d x)}{99 d \sqrt {a \sin (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2856, 2674, 2673} \[ -\frac {16 a^2 \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}-\frac {64 a^3 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}-\frac {2 a \cos ^5(c+d x)}{99 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2673
Rule 2674
Rule 2856
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{11} \int \cos ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {2 a \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{99} (8 a) \int \frac {\cos ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {16 a^2 \cos ^5(c+d x)}{693 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{693} \left (32 a^2\right ) \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {64 a^3 \cos ^5(c+d x)}{3465 d (a+a \sin (c+d x))^{5/2}}-\frac {16 a^2 \cos ^5(c+d x)}{693 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.02, size = 99, normalized size = 0.80 \[ \frac {\sqrt {a (\sin (c+d x)+1)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (-5165 \sin (c+d x)+315 \sin (3 (c+d x))+1960 \cos (2 (c+d x))-3648)}{6930 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 151, normalized size = 1.22 \[ -\frac {2 \, {\left (315 \, \cos \left (d x + c\right )^{6} + 350 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{3} - 16 \, \cos \left (d x + c\right )^{2} + {\left (315 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4} - 40 \, \cos \left (d x + c\right )^{3} - 48 \, \cos \left (d x + c\right )^{2} - 64 \, \cos \left (d x + c\right ) - 128\right )} \sin \left (d x + c\right ) + 64 \, \cos \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3465 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 189, normalized size = 1.52 \[ -\frac {1}{55440} \, \sqrt {2} \sqrt {a} {\left (\frac {385 \, \cos \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {2079 \, \cos \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {6930 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {315 \, \cos \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {1485 \, \cos \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {2310 \, \cos \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.88, size = 75, normalized size = 0.60 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{3} \left (315 \left (\sin ^{3}\left (d x +c \right )\right )+980 \left (\sin ^{2}\left (d x +c \right )\right )+1055 \sin \left (d x +c \right )+422\right )}{3465 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________