Integrand size = 30, antiderivative size = 191 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\frac {-c \cos (d+e x)+b \sin (d+e x)}{5 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{15 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{15 c \left (b^2+c^2\right ) e (c \cos (d+e x)-b \sin (d+e x))} \] Output:
1/5*(-c*cos(e*x+d)+b*sin(e*x+d))/(b^2+c^2)^(1/2)/e/((b^2+c^2)^(1/2)+b*cos( e*x+d)+c*sin(e*x+d))^3-2/15*(c*cos(e*x+d)-b*sin(e*x+d))/(b^2+c^2)/e/((b^2+ c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^2-2/15*(c-(b^2+c^2)^(1/2)*sin(e*x+d) )/c/(b^2+c^2)/e/(c*cos(e*x+d)-b*sin(e*x+d))
Leaf count is larger than twice the leaf count of optimal. \(420\) vs. \(2(191)=382\).
Time = 1.76 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\frac {-76 b^4 c-152 b^2 c^3-76 c^5+90 b c \left (b^2+c^2\right )^{3/2} \cos (d+e x)+20 c \left (-b^4+c^4\right ) \cos (2 (d+e x))+10 b^3 c \sqrt {b^2+c^2} \cos (3 (d+e x))+10 b c^3 \sqrt {b^2+c^2} \cos (3 (d+e x))-4 b^3 c \sqrt {b^2+c^2} \cos (5 (d+e x))+4 b c^3 \sqrt {b^2+c^2} \cos (5 (d+e x))+10 b^4 \sqrt {b^2+c^2} \sin (d+e x)+110 b^2 c^2 \sqrt {b^2+c^2} \sin (d+e x)+100 c^4 \sqrt {b^2+c^2} \sin (d+e x)-40 b^3 c^2 \sin (2 (d+e x))-40 b c^4 \sin (2 (d+e x))-5 b^4 \sqrt {b^2+c^2} \sin (3 (d+e x))+5 c^4 \sqrt {b^2+c^2} \sin (3 (d+e x))+b^4 \sqrt {b^2+c^2} \sin (5 (d+e x))-6 b^2 c^2 \sqrt {b^2+c^2} \sin (5 (d+e x))+c^4 \sqrt {b^2+c^2} \sin (5 (d+e x))}{120 c \left (b^2+c^2\right ) e (c \cos (d+e x)-b \sin (d+e x))^5} \] Input:
Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]
Output:
(-76*b^4*c - 152*b^2*c^3 - 76*c^5 + 90*b*c*(b^2 + c^2)^(3/2)*Cos[d + e*x] + 20*c*(-b^4 + c^4)*Cos[2*(d + e*x)] + 10*b^3*c*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] + 10*b*c^3*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] - 4*b^3*c*Sqrt[b^2 + c^ 2]*Cos[5*(d + e*x)] + 4*b*c^3*Sqrt[b^2 + c^2]*Cos[5*(d + e*x)] + 10*b^4*Sq rt[b^2 + c^2]*Sin[d + e*x] + 110*b^2*c^2*Sqrt[b^2 + c^2]*Sin[d + e*x] + 10 0*c^4*Sqrt[b^2 + c^2]*Sin[d + e*x] - 40*b^3*c^2*Sin[2*(d + e*x)] - 40*b*c^ 4*Sin[2*(d + e*x)] - 5*b^4*Sqrt[b^2 + c^2]*Sin[3*(d + e*x)] + 5*c^4*Sqrt[b ^2 + c^2]*Sin[3*(d + e*x)] + b^4*Sqrt[b^2 + c^2]*Sin[5*(d + e*x)] - 6*b^2* c^2*Sqrt[b^2 + c^2]*Sin[5*(d + e*x)] + c^4*Sqrt[b^2 + c^2]*Sin[5*(d + e*x) ])/(120*c*(b^2 + c^2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x])^5)
Time = 0.59 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3595, 3042, 3595, 3042, 3593}
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.
| \(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}dx\) |
| \(\Big \downarrow \) 3595 |
| \(\displaystyle \frac {2 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2}dx}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2}dx}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\) |
| \(\Big \downarrow \) 3595 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {1}{b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx}{3 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {1}{b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx}{3 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\) |
| \(\Big \downarrow \) 3593 |
| \(\displaystyle \frac {2 \left (-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c-\sqrt {b^2+c^2} \sin (d+e x)}{3 c e \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\) |
Input:
Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]
Output:
-1/5*(c*Cos[d + e*x] - b*Sin[d + e*x])/(Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3) + (2*(-1/3*(c*Cos[d + e*x] - b*Sin[ d + e*x])/(Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c - Sqrt[b^2 + c^2]*Sin[d + e*x])/(3*c*Sqrt[b^2 + c^2]*e*(c*C os[d + e*x] - b*Sin[d + e*x]))))/(5*Sqrt[b^2 + c^2])
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-1), x_Symbol] :> Simp[-(c - a*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[ d + e*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[(c*Cos[d + e*x] - b*Sin[d + e*x])*((a + b*Cos[d + e *x] + c*Sin[d + e*x])^n/(a*e*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1]
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(\frac {4 \left (5 i \sqrt {b^{2}+c^{2}}\, b^{2} c \,{\mathrm e}^{i \left (e x +d \right )}+5 i \sqrt {b^{2}+c^{2}}\, c^{3} {\mathrm e}^{i \left (e x +d \right )}+10 b^{4} {\mathrm e}^{2 i \left (e x +d \right )}+20 b^{2} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+10 c^{4} {\mathrm e}^{2 i \left (e x +d \right )}+2 i b^{3} c +2 i b \,c^{3}+5 \sqrt {b^{2}+c^{2}}\, b^{3} {\mathrm e}^{i \left (e x +d \right )}+5 \sqrt {b^{2}+c^{2}}\, b \,c^{2} {\mathrm e}^{i \left (e x +d \right )}+b^{4}-c^{4}\right ) \left (i b^{3}-3 i b \,c^{2}-3 b^{2} c +c^{3}\right )}{15 e \left (i \sqrt {b^{2}+c^{2}}\, c +b^{2} {\mathrm e}^{i \left (e x +d \right )}+c^{2} {\mathrm e}^{i \left (e x +d \right )}+\sqrt {b^{2}+c^{2}}\, b \right )^{5}}\) | \(246\) |
| derivativedivides | \(\frac {-\frac {2 \left (4 \sqrt {b^{2}+c^{2}}\, b^{2}+\sqrt {b^{2}+c^{2}}\, c^{2}+4 b^{3}+3 b \,c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{2}}-\frac {4 \left (8 \sqrt {b^{2}+c^{2}}\, b^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+8 b^{4}+8 b^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{3}}-\frac {8 \left (24 \sqrt {b^{2}+c^{2}}\, b^{4}+20 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+2 \sqrt {b^{2}+c^{2}}\, c^{4}+24 b^{5}+32 b^{3} c^{2}+9 b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 c^{4}}-\frac {4 \left (48 b^{6}+76 c^{2} b^{4}+31 b^{2} c^{4}+2 c^{6}+48 \sqrt {b^{2}+c^{2}}\, b^{5}+52 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+11 \sqrt {b^{2}+c^{2}}\, b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{3 c^{5}}-\frac {2 \left (192 \sqrt {b^{2}+c^{2}}\, b^{6}+256 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+96 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+7 c^{6} \sqrt {b^{2}+c^{2}}+192 b^{7}+352 b^{5} c^{2}+200 b^{3} c^{4}+35 b \,c^{6}\right )}{15 c^{6}}}{e \,c^{4} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \sqrt {b^{2}+c^{2}}}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right )^{2} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(496\) |
| default | \(\frac {-\frac {2 \left (4 \sqrt {b^{2}+c^{2}}\, b^{2}+\sqrt {b^{2}+c^{2}}\, c^{2}+4 b^{3}+3 b \,c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{2}}-\frac {4 \left (8 \sqrt {b^{2}+c^{2}}\, b^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+8 b^{4}+8 b^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{3}}-\frac {8 \left (24 \sqrt {b^{2}+c^{2}}\, b^{4}+20 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+2 \sqrt {b^{2}+c^{2}}\, c^{4}+24 b^{5}+32 b^{3} c^{2}+9 b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 c^{4}}-\frac {4 \left (48 b^{6}+76 c^{2} b^{4}+31 b^{2} c^{4}+2 c^{6}+48 \sqrt {b^{2}+c^{2}}\, b^{5}+52 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+11 \sqrt {b^{2}+c^{2}}\, b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{3 c^{5}}-\frac {2 \left (192 \sqrt {b^{2}+c^{2}}\, b^{6}+256 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+96 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+7 c^{6} \sqrt {b^{2}+c^{2}}+192 b^{7}+352 b^{5} c^{2}+200 b^{3} c^{4}+35 b \,c^{6}\right )}{15 c^{6}}}{e \,c^{4} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{c}+\frac {2 b \sqrt {b^{2}+c^{2}}}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right )^{2} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(496\) |
Input:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x,method=_RETURNVERBOS E)
Output:
4/15*(5*I*(b^2+c^2)^(1/2)*b^2*c*exp(I*(e*x+d))+5*I*(b^2+c^2)^(1/2)*c^3*exp (I*(e*x+d))+10*b^4*exp(2*I*(e*x+d))+20*b^2*c^2*exp(2*I*(e*x+d))+10*c^4*exp (2*I*(e*x+d))+2*I*b^3*c+2*I*b*c^3+5*(b^2+c^2)^(1/2)*b^3*exp(I*(e*x+d))+5*( b^2+c^2)^(1/2)*b*c^2*exp(I*(e*x+d))+b^4-c^4)*(I*b^3-3*I*b*c^2-3*b^2*c+c^3) /e/(I*(b^2+c^2)^(1/2)*c+b^2*exp(I*(e*x+d))+c^2*exp(I*(e*x+d))+(b^2+c^2)^(1 /2)*b)^5
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (178) = 356\).
Time = 0.12 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {7 \, b^{6} + 26 \, b^{4} c^{2} + 31 \, b^{2} c^{4} + 12 \, c^{6} + 5 \, {\left (b^{6} + b^{4} c^{2} - b^{2} c^{4} - c^{6}\right )} \cos \left (e x + d\right )^{2} + 10 \, {\left (b^{5} c + 2 \, b^{3} c^{3} + b c^{5}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - {\left (2 \, {\left (b^{5} - 10 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \cos \left (e x + d\right )^{5} - 5 \, {\left (b^{5} - 6 \, b^{3} c^{2} + b c^{4}\right )} \cos \left (e x + d\right )^{3} + 5 \, {\left (3 \, b^{5} + 3 \, b^{3} c^{2} + 2 \, b c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, b^{4} c + 25 \, b^{2} c^{3} + 12 \, c^{5} + 2 \, {\left (5 \, b^{4} c - 10 \, b^{2} c^{3} + c^{5}\right )} \cos \left (e x + d\right )^{4} - {\left (15 \, b^{4} c - 10 \, b^{2} c^{3} - c^{5}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}}{15 \, {\left ({\left (5 \, b^{8} c - 14 \, b^{4} c^{5} - 8 \, b^{2} c^{7} + c^{9}\right )} e \cos \left (e x + d\right )^{5} - 10 \, {\left (b^{8} c + b^{6} c^{3} - b^{4} c^{5} - b^{2} c^{7}\right )} e \cos \left (e x + d\right )^{3} + 5 \, {\left (b^{8} c + 2 \, b^{6} c^{3} + b^{4} c^{5}\right )} e \cos \left (e x + d\right ) - {\left ({\left (b^{9} - 8 \, b^{7} c^{2} - 14 \, b^{5} c^{4} + 5 \, b c^{8}\right )} e \cos \left (e x + d\right )^{4} - 2 \, {\left (b^{9} - 3 \, b^{7} c^{2} - 9 \, b^{5} c^{4} - 5 \, b^{3} c^{6}\right )} e \cos \left (e x + d\right )^{2} + {\left (b^{9} + 2 \, b^{7} c^{2} + b^{5} c^{4}\right )} e\right )} \sin \left (e x + d\right )\right )}} \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x, algorithm="fr icas")
Output:
-1/15*(7*b^6 + 26*b^4*c^2 + 31*b^2*c^4 + 12*c^6 + 5*(b^6 + b^4*c^2 - b^2*c ^4 - c^6)*cos(e*x + d)^2 + 10*(b^5*c + 2*b^3*c^3 + b*c^5)*cos(e*x + d)*sin (e*x + d) - (2*(b^5 - 10*b^3*c^2 + 5*b*c^4)*cos(e*x + d)^5 - 5*(b^5 - 6*b^ 3*c^2 + b*c^4)*cos(e*x + d)^3 + 5*(3*b^5 + 3*b^3*c^2 + 2*b*c^4)*cos(e*x + d) + (15*b^4*c + 25*b^2*c^3 + 12*c^5 + 2*(5*b^4*c - 10*b^2*c^3 + c^5)*cos( e*x + d)^4 - (15*b^4*c - 10*b^2*c^3 - c^5)*cos(e*x + d)^2)*sin(e*x + d))*s qrt(b^2 + c^2))/((5*b^8*c - 14*b^4*c^5 - 8*b^2*c^7 + c^9)*e*cos(e*x + d)^5 - 10*(b^8*c + b^6*c^3 - b^4*c^5 - b^2*c^7)*e*cos(e*x + d)^3 + 5*(b^8*c + 2*b^6*c^3 + b^4*c^5)*e*cos(e*x + d) - ((b^9 - 8*b^7*c^2 - 14*b^5*c^4 + 5*b *c^8)*e*cos(e*x + d)^4 - 2*(b^9 - 3*b^7*c^2 - 9*b^5*c^4 - 5*b^3*c^6)*e*cos (e*x + d)^2 + (b^9 + 2*b^7*c^2 + b^5*c^4)*e)*sin(e*x + d))
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/((b**2+c**2)**(1/2)+b*cos(e*x+d)+c*sin(e*x+d))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x, algorithm="ma xima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}\right )}^{3}} \,d x } \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x, algorithm="gi ac")
Output:
sage0*x
Time = 21.31 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.10 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {32\,b^4+32\,b^2\,c^2+4\,c^4}{c^7}+\frac {\left (32\,b^3+16\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^7}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (\frac {8\,b^3+6\,b\,c^2}{c^6}+\frac {\left (8\,b^2+2\,c^2\right )\,\sqrt {b^2+c^2}}{c^6}\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {64\,b^6+\frac {304\,b^4\,c^2}{3}+\frac {124\,b^2\,c^4}{3}+\frac {8\,c^6}{3}}{c^9}+\frac {\sqrt {b^2+c^2}\,\left (64\,b^5+\frac {208\,b^3\,c^2}{3}+\frac {44\,b\,c^4}{3}\right )}{c^9}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {64\,b^5+\frac {256\,b^3\,c^2}{3}+24\,b\,c^4}{c^8}+\frac {\sqrt {b^2+c^2}\,\left (64\,b^4+\frac {160\,b^2\,c^2}{3}+\frac {16\,c^4}{3}\right )}{c^8}\right )+\frac {\frac {128\,b^7}{5}+\frac {704\,b^5\,c^2}{15}+\frac {80\,b^3\,c^4}{3}+\frac {14\,b\,c^6}{3}}{c^{10}}+\frac {\sqrt {b^2+c^2}\,\left (\frac {128\,b^6}{5}+\frac {512\,b^4\,c^2}{15}+\frac {64\,b^2\,c^4}{5}+\frac {14\,c^6}{15}\right )}{c^{10}}}{e\,\left (\frac {16\,b^5+20\,b^3\,c^2+5\,b\,c^4}{c^5}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {40\,b^3+30\,b\,c^2}{c^3}+\frac {\left (40\,b^2+10\,c^2\right )\,\sqrt {b^2+c^2}}{c^3}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {20\,b^2+10\,c^2}{c^2}+\frac {20\,b\,\sqrt {b^2+c^2}}{c^2}\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {40\,b^4+40\,b^2\,c^2+5\,c^4}{c^4}+\frac {\left (40\,b^3+20\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^4}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (\frac {5\,\sqrt {b^2+c^2}}{c}+\frac {5\,b}{c}\right )+\frac {\sqrt {b^2+c^2}\,\left (16\,b^4+12\,b^2\,c^2+c^4\right )}{c^5}\right )} \] Input:
int(1/(b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^3,x)
Output:
-(tan(d/2 + (e*x)/2)^3*((32*b^4 + 4*c^4 + 32*b^2*c^2)/c^7 + ((16*b*c^2 + 3 2*b^3)*(b^2 + c^2)^(1/2))/c^7) + tan(d/2 + (e*x)/2)^4*((6*b*c^2 + 8*b^3)/c ^6 + ((8*b^2 + 2*c^2)*(b^2 + c^2)^(1/2))/c^6) + tan(d/2 + (e*x)/2)*((64*b^ 6 + (8*c^6)/3 + (124*b^2*c^4)/3 + (304*b^4*c^2)/3)/c^9 + ((b^2 + c^2)^(1/2 )*((44*b*c^4)/3 + 64*b^5 + (208*b^3*c^2)/3))/c^9) + tan(d/2 + (e*x)/2)^2*( (24*b*c^4 + 64*b^5 + (256*b^3*c^2)/3)/c^8 + ((b^2 + c^2)^(1/2)*(64*b^4 + ( 16*c^4)/3 + (160*b^2*c^2)/3))/c^8) + ((14*b*c^6)/3 + (128*b^7)/5 + (80*b^3 *c^4)/3 + (704*b^5*c^2)/15)/c^10 + ((b^2 + c^2)^(1/2)*((128*b^6)/5 + (14*c ^6)/15 + (64*b^2*c^4)/5 + (512*b^4*c^2)/15))/c^10)/(e*((5*b*c^4 + 16*b^5 + 20*b^3*c^2)/c^5 + tan(d/2 + (e*x)/2)^2*((30*b*c^2 + 40*b^3)/c^3 + ((40*b^ 2 + 10*c^2)*(b^2 + c^2)^(1/2))/c^3) + tan(d/2 + (e*x)/2)^5 + tan(d/2 + (e* x)/2)^3*((20*b^2 + 10*c^2)/c^2 + (20*b*(b^2 + c^2)^(1/2))/c^2) + tan(d/2 + (e*x)/2)*((40*b^4 + 5*c^4 + 40*b^2*c^2)/c^4 + ((20*b*c^2 + 40*b^3)*(b^2 + c^2)^(1/2))/c^4) + tan(d/2 + (e*x)/2)^4*((5*(b^2 + c^2)^(1/2))/c + (5*b)/ c) + ((b^2 + c^2)^(1/2)*(16*b^4 + c^4 + 12*b^2*c^2))/c^5))
\[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx=\int \frac {1}{\left (\sqrt {b^{2}+c^{2}}+b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{3}}d x \] Input:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x)
Output:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3,x)