Integrand size = 33, antiderivative size = 174 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {4 C x}{a^4}+\frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {4 C \sin (c+d x)}{a^4 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
-4*C*x/a^4+2/105*(3*A+122*C)*sin(d*x+c)/a^4/d+1/105*(3*A-88*C)*cos(d*x+c)^ 2*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2+4*C*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/ 7*(A+C)*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+2/35*(A-6*C)*cos(d*x+ c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(371\) vs. \(2(174)=348\).
Time = 5.91 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.13 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (29400 C d x \cos \left (\frac {d x}{2}\right )+29400 C d x \cos \left (c+\frac {d x}{2}\right )+17640 C d x \cos \left (c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+5880 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+840 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-2520 A \sin \left (\frac {d x}{2}\right )-60830 C \sin \left (\frac {d x}{2}\right )+2520 A \sin \left (c+\frac {d x}{2}\right )+46130 C \sin \left (c+\frac {d x}{2}\right )-1764 A \sin \left (c+\frac {3 d x}{2}\right )-46116 C \sin \left (c+\frac {3 d x}{2}\right )+1260 A \sin \left (2 c+\frac {3 d x}{2}\right )+18060 C \sin \left (2 c+\frac {3 d x}{2}\right )-588 A \sin \left (2 c+\frac {5 d x}{2}\right )-19292 C \sin \left (2 c+\frac {5 d x}{2}\right )+420 A \sin \left (3 c+\frac {5 d x}{2}\right )+2100 C \sin \left (3 c+\frac {5 d x}{2}\right )-144 A \sin \left (3 c+\frac {7 d x}{2}\right )-3791 C \sin \left (3 c+\frac {7 d x}{2}\right )-735 C \sin \left (4 c+\frac {7 d x}{2}\right )-105 C \sin \left (4 c+\frac {9 d x}{2}\right )-105 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{26880 a^4 d} \]
-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*C*d*x*Cos[(d*x)/2] + 29400*C* d*x*Cos[c + (d*x)/2] + 17640*C*d*x*Cos[c + (3*d*x)/2] + 17640*C*d*x*Cos[2* c + (3*d*x)/2] + 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5 *d*x)/2] + 840*C*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 2520*A*Sin[(d*x)/2] - 60830*C*Sin[(d*x)/2] + 2520*A*Sin[c + (d*x)/2] + 46130*C*Sin[c + (d*x)/2] - 1764*A*Sin[c + (3*d*x)/2] - 46116*C*Sin[c + (3* d*x)/2] + 1260*A*Sin[2*c + (3*d*x)/2] + 18060*C*Sin[2*c + (3*d*x)/2] - 588 *A*Sin[2*c + (5*d*x)/2] - 19292*C*Sin[2*c + (5*d*x)/2] + 420*A*Sin[3*c + ( 5*d*x)/2] + 2100*C*Sin[3*c + (5*d*x)/2] - 144*A*Sin[3*c + (7*d*x)/2] - 379 1*C*Sin[3*c + (7*d*x)/2] - 735*C*Sin[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (9 *d*x)/2] - 105*C*Sin[5*c + (9*d*x)/2]))/(a^4*d)
Time = 1.39 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 3521, 3042, 3456, 3042, 3456, 27, 3042, 3447, 3042, 3502, 27, 3042, 3214, 3042, 3127}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (3 A-4 C)+a (A+8 C) \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a (3 A-4 C)+a (A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (6 (A-6 C) a^2+(3 A+52 C) \cos (c+d x) a^2\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (6 (A-6 C) a^2+(3 A+52 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 \cos (c+d x) \left ((3 A-88 C) a^3+(3 A+122 C) \cos (c+d x) a^3\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\cos (c+d x) \left ((3 A-88 C) a^3+(3 A+122 C) \cos (c+d x) a^3\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left ((3 A-88 C) a^3+(3 A+122 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {(3 A+122 C) \cos ^2(c+d x) a^3+(3 A-88 C) \cos (c+d x) a^3}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {(3 A+122 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+(3 A-88 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\int -\frac {210 a^4 C \cos (c+d x)}{\cos (c+d x) a+a}dx}{a}+\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}-210 a^3 C \int \frac {\cos (c+d x)}{\cos (c+d x) a+a}dx\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}-210 a^3 C \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}-210 a^3 C \left (\frac {x}{a}-\int \frac {1}{\cos (c+d x) a+a}dx\right )\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}-210 a^3 C \left (\frac {x}{a}-\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {a^2 (3 A+122 C) \sin (c+d x)}{d}-210 a^3 C \left (\frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)}\right )\right )}{3 a^2}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + (( 2*a*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(1 + Cos[c + d*x])^2) + ( 2*((a^2*(3*A + 122*C)*Sin[c + d*x])/d - 210*a^3*C*(x/a - Sin[c + d*x]/(d*( a + a*Cos[c + d*x])))))/(3*a^2))/(5*a^2))/(7*a^2)
3.1.66.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b* c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Time = 1.84 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {408 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {13 A}{34}+\frac {2741 C}{102}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3 A}{17}+\frac {296 C}{51}\right ) \cos \left (3 d x +3 c \right )+\frac {35 C \cos \left (4 d x +4 c \right )}{136}+\left (A +\frac {3124 C}{51}\right ) \cos \left (d x +c \right )+\frac {A}{2}+\frac {16171 C}{408}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-26880 d x C}{6720 a^{4} d}\) | \(102\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A -\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -16 C \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{8 d \,a^{4}}\) | \(159\) |
| default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A -\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -16 C \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{8 d \,a^{4}}\) | \(159\) |
| risch | \(-\frac {4 C x}{a^{4}}-\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}+\frac {i C \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{4} d}+\frac {2 i \left (105 A \,{\mathrm e}^{6 i \left (d x +c \right )}+1050 C \,{\mathrm e}^{6 i \left (d x +c \right )}+315 A \,{\mathrm e}^{5 i \left (d x +c \right )}+5250 C \,{\mathrm e}^{5 i \left (d x +c \right )}+630 A \,{\mathrm e}^{4 i \left (d x +c \right )}+11900 C \,{\mathrm e}^{4 i \left (d x +c \right )}+630 A \,{\mathrm e}^{3 i \left (d x +c \right )}+14840 C \,{\mathrm e}^{3 i \left (d x +c \right )}+441 A \,{\mathrm e}^{2 i \left (d x +c \right )}+10794 C \,{\mathrm e}^{2 i \left (d x +c \right )}+147 A \,{\mathrm e}^{i \left (d x +c \right )}+4298 C \,{\mathrm e}^{i \left (d x +c \right )}+36 A +764 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(220\) |
| norman | \(\frac {-\frac {4 C x}{a}-\frac {20 C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {20 C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (A -6 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 a d}-\frac {\left (A +C \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (A +65 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (3 A -11 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42 a d}+\frac {\left (3 A +122 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42 a d}+\frac {\left (3 A +226 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (3 A +2075 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {\left (15 A +2542 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42 a d}+\frac {\left (21 A +2059 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{3}}\) | \(342\) |
1/6720*(408*tan(1/2*d*x+1/2*c)*((13/34*A+2741/102*C)*cos(2*d*x+2*c)+(3/17* A+296/51*C)*cos(3*d*x+3*c)+35/136*C*cos(4*d*x+4*c)+(A+3124/51*C)*cos(d*x+c )+1/2*A+16171/408*C)*sec(1/2*d*x+1/2*c)^6-26880*d*x*C)/a^4/d
Time = 0.28 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {420 \, C d x \cos \left (d x + c\right )^{4} + 1680 \, C d x \cos \left (d x + c\right )^{3} + 2520 \, C d x \cos \left (d x + c\right )^{2} + 1680 \, C d x \cos \left (d x + c\right ) + 420 \, C d x - {\left (105 \, C \cos \left (d x + c\right )^{4} + 4 \, {\left (9 \, A + 296 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (6 \, A + 559 \, C\right )} \cos \left (d x + c\right ) + 6 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
-1/105*(420*C*d*x*cos(d*x + c)^4 + 1680*C*d*x*cos(d*x + c)^3 + 2520*C*d*x* cos(d*x + c)^2 + 1680*C*d*x*cos(d*x + c) + 420*C*d*x - (105*C*cos(d*x + c) ^4 + 4*(9*A + 296*C)*cos(d*x + c)^3 + (39*A + 2636*C)*cos(d*x + c)^2 + 4*( 6*A + 559*C)*cos(d*x + c) + 6*A + 664*C)*sin(d*x + c))/(a^4*d*cos(d*x + c) ^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c ) + a^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (168) = 336\).
Time = 4.82 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} - \frac {15 A \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {48 A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {42 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {105 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 C \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{3}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Piecewise((-15*A*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840 *a**4*d) + 48*A*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840* a**4*d) - 42*A*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a **4*d) + 105*A*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4 *d) - 3360*C*d*x*tan(c/2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840 *a**4*d) - 3360*C*d*x/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 15*C *tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 132*C *tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 658*C *tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 4340* C*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825 *C*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d), Ne(d, 0 )), (x*(A + C*cos(c)**2)*cos(c)**3/(a*cos(c) + a)**4, True))
Time = 0.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
1/840*(C*(1680*sin(d*x + c)/((a^4 + a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^ 2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/(cos(d*x + c) + 1) - 805*sin(d *x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(c os(d*x + c) + 1))/a^4) + 3*A*(35*sin(d*x + c)/(cos(d*x + c) + 1) - 35*sin( d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4)/d
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {3360 \, {\left (d x + c\right )} C}{a^{4}} - \frac {1680 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
-1/840*(3360*(d*x + c)*C/a^4 - 1680*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^4) + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/ 2*d*x + 1/2*c)^7 - 63*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 147*C*a^24*tan(1/2*d *x + 1/2*c)^5 + 105*A*a^24*tan(1/2*d*x + 1/2*c)^3 + 805*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 105*A*a^24*tan(1/2*d*x + 1/2*c) - 5145*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 1.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {2\,C\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d}-\frac {\left (-\frac {12\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {764\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {23\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}+\frac {143\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {9\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {8\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {4\,C\,x}{a^4} \]
(2*C*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2))/(a^4*d) - ((A*sin(c/2 + (d*x)/ 2))/56 + (C*sin(c/2 + (d*x)/2))/56 - cos(c/2 + (d*x)/2)^2*((9*A*sin(c/2 + (d*x)/2))/70 + (8*C*sin(c/2 + (d*x)/2))/35) + cos(c/2 + (d*x)/2)^4*((23*A* sin(c/2 + (d*x)/2))/70 + (143*C*sin(c/2 + (d*x)/2))/105) - cos(c/2 + (d*x) /2)^6*((12*A*sin(c/2 + (d*x)/2))/35 + (764*C*sin(c/2 + (d*x)/2))/105))/(a^ 4*d*cos(c/2 + (d*x)/2)^7) - (4*C*x)/a^4