Integrand size = 12, antiderivative size = 87 \[ \int (a+a \sin (c+d x))^4 \, dx=\frac {35 a^4 x}{8}-\frac {8 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {27 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
35/8*a^4*x-8*a^4*cos(d*x+c)/d+4/3*a^4*cos(d*x+c)^3/d-27/8*a^4*cos(d*x+c)*s in(d*x+c)/d-1/4*a^4*cos(d*x+c)*sin(d*x+c)^3/d
Time = 3.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (-672 \cos (c+d x)+32 \cos (3 (c+d x))+3 (140 c+140 d x-56 \sin (2 (c+d x))+\sin (4 (c+d x))))}{96 d} \]
(a^4*(-672*Cos[c + d*x] + 32*Cos[3*(c + d*x)] + 3*(140*c + 140*d*x - 56*Si n[2*(c + d*x)] + Sin[4*(c + d*x)])))/(96*d)
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3124, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a \sin (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (c+d x)+a)^4dx\) |
\(\Big \downarrow \) 3124 |
\(\displaystyle \int \left (a^4 \sin ^4(c+d x)+4 a^4 \sin ^3(c+d x)+6 a^4 \sin ^2(c+d x)+4 a^4 \sin (c+d x)+a^4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {8 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\) |
(35*a^4*x)/8 - (8*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (27 *a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/ (4*d)
3.1.40.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Time = 1.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(-\frac {a^{4} \left (-420 d x +672 \cos \left (d x +c \right )-3 \sin \left (4 d x +4 c \right )-32 \cos \left (3 d x +3 c \right )+168 \sin \left (2 d x +2 c \right )+640\right )}{96 d}\) | \(56\) |
risch | \(\frac {35 a^{4} x}{8}-\frac {7 a^{4} \cos \left (d x +c \right )}{d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}-\frac {7 a^{4} \sin \left (2 d x +2 c \right )}{4 d}\) | \(73\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-4 a^{4} \cos \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-4 a^{4} \cos \left (d x +c \right )+a^{4} \left (d x +c \right )}{d}\) | \(111\) |
parts | \(a^{4} x +\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}-\frac {4 a^{4} \cos \left (d x +c \right )}{d}+\frac {6 a^{4} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3 d}\) | \(115\) |
norman | \(\frac {\frac {35 a^{4} x}{8}-\frac {40 a^{4}}{3 d}-\frac {27 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {35 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {27 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {105 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {35 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {8 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {40 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {136 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(231\) |
-1/96*a^4*(-420*d*x+672*cos(d*x+c)-3*sin(4*d*x+4*c)-32*cos(3*d*x+3*c)+168* sin(2*d*x+2*c)+640)/d
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int (a+a \sin (c+d x))^4 \, dx=\frac {32 \, a^{4} \cos \left (d x + c\right )^{3} + 105 \, a^{4} d x - 192 \, a^{4} \cos \left (d x + c\right ) + 3 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{3} - 29 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(32*a^4*cos(d*x + c)^3 + 105*a^4*d*x - 192*a^4*cos(d*x + c) + 3*(2*a^ 4*cos(d*x + c)^3 - 29*a^4*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (82) = 164\).
Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.57 \[ \int (a+a \sin (c+d x))^4 \, dx=\begin {cases} \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 a^{4} x \cos ^{2}{\left (c + d x \right )} + a^{4} x - \frac {5 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {8 a^{4} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{4} \cos {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((3*a**4*x*sin(c + d*x)**4/8 + 3*a**4*x*sin(c + d*x)**2*cos(c + d *x)**2/4 + 3*a**4*x*sin(c + d*x)**2 + 3*a**4*x*cos(c + d*x)**4/8 + 3*a**4* x*cos(c + d*x)**2 + a**4*x - 5*a**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) - 4 *a**4*sin(c + d*x)**2*cos(c + d*x)/d - 3*a**4*sin(c + d*x)*cos(c + d*x)**3 /(8*d) - 3*a**4*sin(c + d*x)*cos(c + d*x)/d - 8*a**4*cos(c + d*x)**3/(3*d) - 4*a**4*cos(c + d*x)/d, Ne(d, 0)), (x*(a*sin(c) + a)**4, True))
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.24 \[ \int (a+a \sin (c+d x))^4 \, dx=a^{4} x + \frac {4 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {3 \, {\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{2 \, d} - \frac {4 \, a^{4} \cos \left (d x + c\right )}{d} \]
a^4*x + 4/3*(cos(d*x + c)^3 - 3*cos(d*x + c))*a^4/d + 1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) - 8*sin(2*d*x + 2*c))*a^4/d + 3/2*(2*d*x + 2*c - sin(2* d*x + 2*c))*a^4/d - 4*a^4*cos(d*x + c)/d
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int (a+a \sin (c+d x))^4 \, dx=\frac {35}{8} \, a^{4} x + \frac {a^{4} \cos \left (3 \, d x + 3 \, c\right )}{3 \, d} - \frac {7 \, a^{4} \cos \left (d x + c\right )}{d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
35/8*a^4*x + 1/3*a^4*cos(3*d*x + 3*c)/d - 7*a^4*cos(d*x + c)/d + 1/32*a^4* sin(4*d*x + 4*c)/d - 7/4*a^4*sin(2*d*x + 2*c)/d
Time = 8.15 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.72 \[ \int (a+a \sin (c+d x))^4 \, dx=\frac {35\,a^4\,x}{8}-\frac {\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {27\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {a^4\,\left (105\,c+105\,d\,x\right )}{24}-\frac {a^4\,\left (105\,c+105\,d\,x-320\right )}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{6}-\frac {a^4\,\left (420\,c+420\,d\,x-192\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{6}-\frac {a^4\,\left (420\,c+420\,d\,x-1088\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (105\,c+105\,d\,x\right )}{4}-\frac {a^4\,\left (630\,c+630\,d\,x-960\right )}{24}\right )+\frac {27\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
(35*a^4*x)/8 - ((35*a^4*tan(c/2 + (d*x)/2)^3)/4 - (35*a^4*tan(c/2 + (d*x)/ 2)^5)/4 - (27*a^4*tan(c/2 + (d*x)/2)^7)/4 + (a^4*(105*c + 105*d*x))/24 - ( a^4*(105*c + 105*d*x - 320))/24 + tan(c/2 + (d*x)/2)^6*((a^4*(105*c + 105* d*x))/6 - (a^4*(420*c + 420*d*x - 192))/24) + tan(c/2 + (d*x)/2)^2*((a^4*( 105*c + 105*d*x))/6 - (a^4*(420*c + 420*d*x - 1088))/24) + tan(c/2 + (d*x) /2)^4*((a^4*(105*c + 105*d*x))/4 - (a^4*(630*c + 630*d*x - 960))/24) + (27 *a^4*tan(c/2 + (d*x)/2))/4)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^4)