Integrand size = 29, antiderivative size = 87 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7 x}{8 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d} \]
7/8*x/a^2+2*cos(d*x+c)/a^2/d-2/3*cos(d*x+c)^3/a^2/d-7/8*cos(d*x+c)*sin(d*x +c)/a^2/d-1/4*cos(d*x+c)*sin(d*x+c)^3/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(87)=174\).
Time = 0.91 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.97 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {168 d x \cos \left (\frac {c}{2}\right )+144 \cos \left (\frac {c}{2}+d x\right )+144 \cos \left (\frac {3 c}{2}+d x\right )-48 \cos \left (\frac {3 c}{2}+2 d x\right )+48 \cos \left (\frac {5 c}{2}+2 d x\right )-16 \cos \left (\frac {5 c}{2}+3 d x\right )-16 \cos \left (\frac {7 c}{2}+3 d x\right )+3 \cos \left (\frac {7 c}{2}+4 d x\right )-3 \cos \left (\frac {9 c}{2}+4 d x\right )+8 \sin \left (\frac {c}{2}\right )+168 d x \sin \left (\frac {c}{2}\right )-144 \sin \left (\frac {c}{2}+d x\right )+144 \sin \left (\frac {3 c}{2}+d x\right )-48 \sin \left (\frac {3 c}{2}+2 d x\right )-48 \sin \left (\frac {5 c}{2}+2 d x\right )+16 \sin \left (\frac {5 c}{2}+3 d x\right )-16 \sin \left (\frac {7 c}{2}+3 d x\right )+3 \sin \left (\frac {7 c}{2}+4 d x\right )+3 \sin \left (\frac {9 c}{2}+4 d x\right )}{192 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
(168*d*x*Cos[c/2] + 144*Cos[c/2 + d*x] + 144*Cos[(3*c)/2 + d*x] - 48*Cos[( 3*c)/2 + 2*d*x] + 48*Cos[(5*c)/2 + 2*d*x] - 16*Cos[(5*c)/2 + 3*d*x] - 16*C os[(7*c)/2 + 3*d*x] + 3*Cos[(7*c)/2 + 4*d*x] - 3*Cos[(9*c)/2 + 4*d*x] + 8* Sin[c/2] + 168*d*x*Sin[c/2] - 144*Sin[c/2 + d*x] + 144*Sin[(3*c)/2 + d*x] - 48*Sin[(3*c)/2 + 2*d*x] - 48*Sin[(5*c)/2 + 2*d*x] + 16*Sin[(5*c)/2 + 3*d *x] - 16*Sin[(7*c)/2 + 3*d*x] + 3*Sin[(7*c)/2 + 4*d*x] + 3*Sin[(9*c)/2 + 4 *d*x])/(192*a^2*d*(Cos[c/2] + Sin[c/2]))
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle \frac {\int \sin ^2(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin (c+d x)^2 (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \frac {\int \left (a^2 \sin ^4(c+d x)-2 a^2 \sin ^3(c+d x)+a^2 \sin ^2(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {7 a^2 x}{8}}{a^4}\) |
((7*a^2*x)/8 + (2*a^2*Cos[c + d*x])/d - (2*a^2*Cos[c + d*x]^3)/(3*d) - (7* a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/( 4*d))/a^4
3.5.23.3.1 Defintions of rubi rules used
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
| method | result | size |
| parallelrisch | \(\frac {84 d x -16 \cos \left (3 d x +3 c \right )+144 \cos \left (d x +c \right )+3 \sin \left (4 d x +4 c \right )-48 \sin \left (2 d x +2 c \right )+128}{96 d \,a^{2}}\) | \(56\) |
| risch | \(\frac {7 x}{8 a^{2}}+\frac {3 \cos \left (d x +c \right )}{2 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{6 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(73\) |
| derivativedivides | \(\frac {\frac {8 \left (\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {15 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(114\) |
| default | \(\frac {\frac {8 \left (\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {15 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(114\) |
| norman | \(\frac {\frac {357 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {133 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {525 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {63 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {455 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {231 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {525 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {455 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {357 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {231 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {8}{3 a d}+\frac {133 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {63 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {7 x}{8 a}+\frac {63 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {75 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {21 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {7 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {21 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {101 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {21 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {893 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {229 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {99 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {339 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {1187 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {113 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {205 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {25 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(565\) |
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {16 \, \cos \left (d x + c\right )^{3} - 21 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 48 \, \cos \left (d x + c\right )}{24 \, a^{2} d} \]
-1/24*(16*cos(d*x + c)^3 - 21*d*x - 3*(2*cos(d*x + c)^3 - 9*cos(d*x + c))* sin(d*x + c) - 48*cos(d*x + c))/(a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (82) = 164\).
Time = 20.55 (sec) , antiderivative size = 1153, normalized size of antiderivative = 13.25 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((21*d*x*tan(c/2 + d*x/2)**8/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96* a**2*d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*ta n(c/2 + d*x/2)**2 + 24*a**2*d) + 84*d*x*tan(c/2 + d*x/2)**6/(24*a**2*d*tan (c/2 + d*x/2)**8 + 96*a**2*d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d* x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d) + 126*d*x*tan(c/2 + d *x/2)**4/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96*a**2*d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d ) + 84*d*x*tan(c/2 + d*x/2)**2/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96*a**2*d* tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d) + 21*d*x/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96*a**2* d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d) + 42*tan(c/2 + d*x/2)**7/(24*a**2*d*tan(c/2 + d* x/2)**8 + 96*a**2*d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d) + 90*tan(c/2 + d*x/2)**5/(24*a **2*d*tan(c/2 + d*x/2)**8 + 96*a**2*d*tan(c/2 + d*x/2)**6 + 144*a**2*d*tan (c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a**2*d) + 192*tan(c/ 2 + d*x/2)**4/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96*a**2*d*tan(c/2 + d*x/2)* *6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c/2 + d*x/2)**2 + 24*a **2*d) - 90*tan(c/2 + d*x/2)**3/(24*a**2*d*tan(c/2 + d*x/2)**8 + 96*a**2*d *tan(c/2 + d*x/2)**6 + 144*a**2*d*tan(c/2 + d*x/2)**4 + 96*a**2*d*tan(c...
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (79) = 158\).
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {128 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {21 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 32}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {21 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{12 \, d} \]
-1/12*((21*sin(d*x + c)/(cos(d*x + c) + 1) - 128*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 45*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 96*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 - 45*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 21*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 32)/(a^2 + 4*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^2*sin(d*x + c)^6 /(cos(d*x + c) + 1)^6 + a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 21*arct an(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.70 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {21 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 128 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \]
1/24*(21*(d*x + c)/a^2 + 2*(21*tan(1/2*d*x + 1/2*c)^7 + 45*tan(1/2*d*x + 1 /2*c)^5 + 96*tan(1/2*d*x + 1/2*c)^4 - 45*tan(1/2*d*x + 1/2*c)^3 + 128*tan( 1/2*d*x + 1/2*c)^2 - 21*tan(1/2*d*x + 1/2*c) + 32)/((tan(1/2*d*x + 1/2*c)^ 2 + 1)^4*a^2))/d
Time = 9.90 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7\,x}{8\,a^2}+\frac {2\,\cos \left (c+d\,x\right )}{a^2\,d}-\frac {2\,{\cos \left (c+d\,x\right )}^3}{3\,a^2\,d}+\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^2\,d}-\frac {9\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^2\,d} \]