Integrand size = 30, antiderivative size = 259 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\frac {-c \cos (d+e x)+b \sin (d+e x)}{7 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{35 \left (b^2+c^2\right )^{3/2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{35 c \left (b^2+c^2\right )^{3/2} e (c \cos (d+e x)-b \sin (d+e x))} \] Output:
1/7*(-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))^4-3/35*(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))^3-2/35*(c*cos(e*x+d)-b*sin(e*x+d))/( b^2+c^2)^(3/2)/e/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^2-2/35*(c-(b^ 2+c^2)^(1/2)*sin(e*x+d))/c/(b^2+c^2)^(3/2)/e/(c*cos(e*x+d)-b*sin(e*x+d))
Leaf count is larger than twice the leaf count of optimal. \(533\) vs. \(2(259)=518\).
Time = 1.86 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.06 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\frac {832 b^4 c \sqrt {b^2+c^2}+1664 b^2 c^3 \sqrt {b^2+c^2}+832 c^5 \sqrt {b^2+c^2}-1190 b c \left (b^2+c^2\right )^2 \cos (d+e x)+448 c \sqrt {b^2+c^2} \left (b^4-c^4\right ) \cos (2 (d+e x))-112 b^5 c \cos (3 (d+e x))+56 b^3 c^3 \cos (3 (d+e x))+168 b c^5 \cos (3 (d+e x))+28 b^5 c \cos (5 (d+e x))-28 b c^5 \cos (5 (d+e x))-6 b^5 c \cos (7 (d+e x))+20 b^3 c^3 \cos (7 (d+e x))-6 b c^5 \cos (7 (d+e x))-35 b^6 \sin (d+e x)-1295 b^4 c^2 \sin (d+e x)-2485 b^2 c^4 \sin (d+e x)-1225 c^6 \sin (d+e x)+896 b^3 c^2 \sqrt {b^2+c^2} \sin (2 (d+e x))+896 b c^4 \sqrt {b^2+c^2} \sin (2 (d+e x))+21 b^6 \sin (3 (d+e x))-189 b^4 c^2 \sin (3 (d+e x))-161 b^2 c^4 \sin (3 (d+e x))+49 c^6 \sin (3 (d+e x))-7 b^6 \sin (5 (d+e x))+35 b^4 c^2 \sin (5 (d+e x))+35 b^2 c^4 \sin (5 (d+e x))-7 c^6 \sin (5 (d+e x))+b^6 \sin (7 (d+e x))-15 b^4 c^2 \sin (7 (d+e x))+15 b^2 c^4 \sin (7 (d+e x))-c^6 \sin (7 (d+e x))}{1120 c \left (b^2+c^2\right ) e (-c \cos (d+e x)+b \sin (d+e x))^7} \] Input:
Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]
Output:
(832*b^4*c*Sqrt[b^2 + c^2] + 1664*b^2*c^3*Sqrt[b^2 + c^2] + 832*c^5*Sqrt[b ^2 + c^2] - 1190*b*c*(b^2 + c^2)^2*Cos[d + e*x] + 448*c*Sqrt[b^2 + c^2]*(b ^4 - c^4)*Cos[2*(d + e*x)] - 112*b^5*c*Cos[3*(d + e*x)] + 56*b^3*c^3*Cos[3 *(d + e*x)] + 168*b*c^5*Cos[3*(d + e*x)] + 28*b^5*c*Cos[5*(d + e*x)] - 28* b*c^5*Cos[5*(d + e*x)] - 6*b^5*c*Cos[7*(d + e*x)] + 20*b^3*c^3*Cos[7*(d + e*x)] - 6*b*c^5*Cos[7*(d + e*x)] - 35*b^6*Sin[d + e*x] - 1295*b^4*c^2*Sin[ d + e*x] - 2485*b^2*c^4*Sin[d + e*x] - 1225*c^6*Sin[d + e*x] + 896*b^3*c^2 *Sqrt[b^2 + c^2]*Sin[2*(d + e*x)] + 896*b*c^4*Sqrt[b^2 + c^2]*Sin[2*(d + e *x)] + 21*b^6*Sin[3*(d + e*x)] - 189*b^4*c^2*Sin[3*(d + e*x)] - 161*b^2*c^ 4*Sin[3*(d + e*x)] + 49*c^6*Sin[3*(d + e*x)] - 7*b^6*Sin[5*(d + e*x)] + 35 *b^4*c^2*Sin[5*(d + e*x)] + 35*b^2*c^4*Sin[5*(d + e*x)] - 7*c^6*Sin[5*(d + e*x)] + b^6*Sin[7*(d + e*x)] - 15*b^4*c^2*Sin[7*(d + e*x)] + 15*b^2*c^4*S in[7*(d + e*x)] - c^6*Sin[7*(d + e*x)])/(1120*c*(b^2 + c^2)*e*(-(c*Cos[d + e*x]) + b*Sin[d + e*x])^7)
Time = 0.79 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3595, 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.
| \(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}dx\) |
| \(\Big \downarrow \) 3595 |
| \(\displaystyle \frac {3 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^3}dx}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^3}dx}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3595 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2}dx}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\right )}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2}dx}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\right )}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3595 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx}{3 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\right )}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx}{3 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\right )}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
| \(\Big \downarrow \) 3593 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c-\sqrt {b^2+c^2} \sin (d+e x)}{3 c e \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}\right )}{5 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{5 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}\right )}{7 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}\) |
Input:
Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]
Output:
-1/7*(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])^4) + (3*(-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*Cos[d + e*x] - b*Sin[d + e*x]) )))/(5*Sqrt[b^2 + c^2])))/(7*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 = 2.18 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.97
| method | result | size |
| risch | \(\frac {4 \left (14 i b \,c^{5} {\mathrm e}^{i \left (e x +d \right )}+21 i \sqrt {b^{2}+c^{2}}\, c^{5} {\mathrm e}^{2 i \left (e x +d \right )}+28 i b^{3} c^{3} {\mathrm e}^{i \left (e x +d \right )}+35 b^{6} {\mathrm e}^{3 i \left (e x +d \right )}+105 b^{4} c^{2} {\mathrm e}^{3 i \left (e x +d \right )}+105 b^{2} c^{4} {\mathrm e}^{3 i \left (e x +d \right )}+35 c^{6} {\mathrm e}^{3 i \left (e x +d \right )}+21 \sqrt {b^{2}+c^{2}}\, b^{5} {\mathrm e}^{2 i \left (e x +d \right )}+42 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+21 \sqrt {b^{2}+c^{2}}\, b \,c^{4} {\mathrm e}^{2 i \left (e x +d \right )}-i \sqrt {b^{2}+c^{2}}\, c^{5}+21 i \sqrt {b^{2}+c^{2}}\, b^{4} c \,{\mathrm e}^{2 i \left (e x +d \right )}+2 i \sqrt {b^{2}+c^{2}}\, b^{2} c^{3}+42 i \sqrt {b^{2}+c^{2}}\, b^{2} c^{3} {\mathrm e}^{2 i \left (e x +d \right )}+3 i \sqrt {b^{2}+c^{2}}\, b^{4} c +14 i b^{5} c \,{\mathrm e}^{i \left (e x +d \right )}+7 b^{6} {\mathrm e}^{i \left (e x +d \right )}+7 b^{4} c^{2} {\mathrm e}^{i \left (e x +d \right )}-7 b^{2} c^{4} {\mathrm e}^{i \left (e x +d \right )}-7 c^{6} {\mathrm e}^{i \left (e x +d \right )}+\sqrt {b^{2}+c^{2}}\, b^{5}-2 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}-3 \sqrt {b^{2}+c^{2}}\, b \,c^{4}\right ) \left (i b^{4}-6 i b^{2} c^{2}+i c^{4}-4 b^{3} c +4 c^{3} b \right )}{35 \left (i \sqrt {b^{2}+c^{2}}\, c +b^{2} {\mathrm e}^{i \left (e x +d \right )}+c^{2} {\mathrm e}^{i \left (e x +d \right )}+\sqrt {b^{2}+c^{2}}\, b \right )^{7} e}\) | \(511\) |
| derivativedivides | \(-\frac {2 \left (\frac {\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 )^{6}}{c^{2}}+\frac {3 \left (16 \sqrt {b^{2}+c^{2}}\, b^{4}+12 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+\sqrt {b^{2}+c^{2}}\, c^{4}+16 b^{5}+20 b^{3} c^{2}+5 b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{c^{3}}+\frac {2 \left (80 \sqrt {b^{2}+c^{2}}\, b^{5}+84 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+17 \sqrt {b^{2}+c^{2}}\, b \,c^{4}+80 b^{6}+124 c^{2} b^{4}+49 b^{2} c^{4}+3 c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{4}}+\frac {2 \left (160 b^{7}+288 b^{5} c^{2}+150 b^{3} c^{4}+20 b \,c^{6}+160 \sqrt {b^{2}+c^{2}}\, b^{6}+208 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+66 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+3 c^{6} \sqrt {b^{2}+c^{2}}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{5}}+\frac {3 \left (640 b^{7} \sqrt {b^{2}+c^{2}}+992 \sqrt {b^{2}+c^{2}}\, b^{5} c^{2}+440 b^{3} \sqrt {b^{2}+c^{2}}\, c^{4}+50 b \,c^{6} \sqrt {b^{2}+c^{2}}+640 b^{8}+1312 b^{6} c^{2}+856 b^{4} c^{4}+186 b^{2} c^{6}+7 c^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{5 c^{6}}+\frac {\left (1280 b^{9}+2944 b^{7} c^{2}+2288 b^{5} c^{4}+676 b^{3} c^{6}+57 b \,c^{8}+1280 \sqrt {b^{2}+c^{2}}\, b^{8}+2304 \sqrt {b^{2}+c^{2}}\, b^{6} c^{2}+1296 \sqrt {b^{2}+c^{2}}\, b^{4} c^{4}+236 \sqrt {b^{2}+c^{2}}\, b^{2} c^{6}+7 \sqrt {b^{2}+c^{2}}\, c^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{5 c^{7}}+\frac {\frac {512 \sqrt {b^{2}+c^{2}}\, b^{9}}{7}+\frac {5248 \sqrt {b^{2}+c^{2}}\, b^{7} c^{2}}{35}+\frac {512 \sqrt {b^{2}+c^{2}}\, b^{5} c^{4}}{5}+\frac {136 \sqrt {b^{2}+c^{2}}\, b^{3} c^{6}}{5}+\frac {12 \sqrt {b^{2}+c^{2}}\, b \,c^{8}}{5}+\frac {512 b^{10}}{7}+\frac {6528 b^{8} c^{2}}{35}+\frac {5888 b^{6} c^{4}}{35}+\frac {2248 b^{4} c^{6}}{35}+\frac {68 b^{2} c^{8}}{7}+\frac {12 c^{10}}{35}}{c^{8}}\right )}{e \,c^{6} \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 )^{3} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(823\) |
| default | \(-\frac {2 \left (\frac {\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 )^{6}}{c^{2}}+\frac {3 \left (16 \sqrt {b^{2}+c^{2}}\, b^{4}+12 \sqrt {b^{2}+c^{2}}\, b^{2} c^{2}+\sqrt {b^{2}+c^{2}}\, c^{4}+16 b^{5}+20 b^{3} c^{2}+5 b \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{c^{3}}+\frac {2 \left (80 \sqrt {b^{2}+c^{2}}\, b^{5}+84 \sqrt {b^{2}+c^{2}}\, b^{3} c^{2}+17 \sqrt {b^{2}+c^{2}}\, b \,c^{4}+80 b^{6}+124 c^{2} b^{4}+49 b^{2} c^{4}+3 c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{c^{4}}+\frac {2 \left (160 b^{7}+288 b^{5} c^{2}+150 b^{3} c^{4}+20 b \,c^{6}+160 \sqrt {b^{2}+c^{2}}\, b^{6}+208 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+66 b^{2} \sqrt {b^{2}+c^{2}}\, c^{4}+3 c^{6} \sqrt {b^{2}+c^{2}}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{c^{5}}+\frac {3 \left (640 b^{7} \sqrt {b^{2}+c^{2}}+992 \sqrt {b^{2}+c^{2}}\, b^{5} c^{2}+440 b^{3} \sqrt {b^{2}+c^{2}}\, c^{4}+50 b \,c^{6} \sqrt {b^{2}+c^{2}}+640 b^{8}+1312 b^{6} c^{2}+856 b^{4} c^{4}+186 b^{2} c^{6}+7 c^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{5 c^{6}}+\frac {\left (1280 b^{9}+2944 b^{7} c^{2}+2288 b^{5} c^{4}+676 b^{3} c^{6}+57 b \,c^{8}+1280 \sqrt {b^{2}+c^{2}}\, b^{8}+2304 \sqrt {b^{2}+c^{2}}\, b^{6} c^{2}+1296 \sqrt {b^{2}+c^{2}}\, b^{4} c^{4}+236 \sqrt {b^{2}+c^{2}}\, b^{2} c^{6}+7 \sqrt {b^{2}+c^{2}}\, c^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{5 c^{7}}+\frac {\frac {512 \sqrt {b^{2}+c^{2}}\, b^{9}}{7}+\frac {5248 \sqrt {b^{2}+c^{2}}\, b^{7} c^{2}}{35}+\frac {512 \sqrt {b^{2}+c^{2}}\, b^{5} c^{4}}{5}+\frac {136 \sqrt {b^{2}+c^{2}}\, b^{3} c^{6}}{5}+\frac {12 \sqrt {b^{2}+c^{2}}\, b \,c^{8}}{5}+\frac {512 b^{10}}{7}+\frac {6528 b^{8} c^{2}}{35}+\frac {5888 b^{6} c^{4}}{35}+\frac {2248 b^{4} c^{6}}{35}+\frac {68 b^{2} c^{8}}{7}+\frac {12 c^{10}}{35}}{c^{8}}\right )}{e \,c^{6} \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 )^{3} \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )}\) | \(823\) |
Input:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x,method=_RETURNVERBOS E)
Output:
4/35*(14*I*b*c^5*exp(I*(e*x+d))+21*I*(b^2+c^2)^(1/2)*c^5*exp(2*I*(e*x+d))+ 28*I*b^3*c^3*exp(I*(e*x+d))+35*b^6*exp(3*I*(e*x+d))+105*b^4*c^2*exp(3*I*(e *x+d))+105*b^2*c^4*exp(3*I*(e*x+d))+35*c^6*exp(3*I*(e*x+d))+21*(b^2+c^2)^( 1/2)*b^5*exp(2*I*(e*x+d))+42*(b^2+c^2)^(1/2)*b^3*c^2*exp(2*I*(e*x+d))+21*( b^2+c^2)^(1/2)*b*c^4*exp(2*I*(e*x+d))-I*(b^2+c^2)^(1/2)*c^5+21*I*(b^2+c^2) ^(1/2)*b^4*c*exp(2*I*(e*x+d))+2*I*(b^2+c^2)^(1/2)*b^2*c^3+42*I*(b^2+c^2)^( 1/2)*b^2*c^3*exp(2*I*(e*x+d))+3*I*(b^2+c^2)^(1/2)*b^4*c+14*I*b^5*c*exp(I*( e*x+d))+7*b^6*exp(I*(e*x+d))+7*b^4*c^2*exp(I*(e*x+d))-7*b^2*c^4*exp(I*(e*x +d))-7*c^6*exp(I*(e*x+d))+(b^2+c^2)^(1/2)*b^5-2*(b^2+c^2)^(1/2)*b^3*c^2-3* (b^2+c^2)^(1/2)*b*c^4)*(I*b^4-6*I*b^2*c^2+I*c^4-4*b^3*c+4*c^3*b)/(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)^7/e
Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (238) = 476\).
Time = 0.28 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="fr icas")
Output:
1/35*(2*(b^7 - 21*b^5*c^2 + 35*b^3*c^4 - 7*b*c^6)*cos(e*x + d)^7 - 7*(b^7 - 15*b^5*c^2 + 15*b^3*c^4 - b*c^6)*cos(e*x + d)^5 - 14*(5*b^5*c^2 - 5*b^3* c^4 - 2*b*c^6)*cos(e*x + d)^3 - 7*(5*b^7 + 15*b^5*c^2 + 20*b^3*c^4 + 8*b*c ^6)*cos(e*x + d) - (35*b^6*c + 105*b^4*c^3 + 112*b^2*c^5 + 40*c^7 - 2*(7*b ^6*c - 35*b^4*c^3 + 21*b^2*c^5 - c^7)*cos(e*x + d)^6 + (35*b^6*c - 105*b^4 *c^3 + 21*b^2*c^5 + c^7)*cos(e*x + d)^4 + 2*(35*b^4*c^3 + 7*b^2*c^5 - 4*c^ 7)*cos(e*x + d)^2)*sin(e*x + d) + 4*(3*b^6 + 16*b^4*c^2 + 23*b^2*c^4 + 10* c^6 + 7*(b^6 + b^4*c^2 - b^2*c^4 - c^6)*cos(e*x + d)^2 + 14*(b^5*c + 2*b^3 *c^3 + b*c^5)*cos(e*x + d)*sin(e*x + d))*sqrt(b^2 + c^2))/((7*b^10*c - 21* b^8*c^3 - 42*b^6*c^5 + 6*b^4*c^7 + 19*b^2*c^9 - c^11)*e*cos(e*x + d)^7 - 7 *(3*b^10*c - 4*b^8*c^3 - 14*b^6*c^5 - 4*b^4*c^7 + 3*b^2*c^9)*e*cos(e*x + d )^5 + 7*(3*b^10*c + b^8*c^3 - 7*b^6*c^5 - 5*b^4*c^7)*e*cos(e*x + d)^3 - 7* (b^10*c + 2*b^8*c^3 + b^6*c^5)*e*cos(e*x + d) - ((b^11 - 19*b^9*c^2 - 6*b^ 7*c^4 + 42*b^5*c^6 + 21*b^3*c^8 - 7*b*c^10)*e*cos(e*x + d)^6 - (3*b^11 - 3 6*b^9*c^2 - 46*b^7*c^4 + 28*b^5*c^6 + 35*b^3*c^8)*e*cos(e*x + d)^4 + 3*(b^ 11 - 5*b^9*c^2 - 13*b^7*c^4 - 7*b^5*c^6)*e*cos(e*x + d)^2 - (b^11 + 2*b^9* c^2 + b^7*c^4)*e)*sin(e*x + d))
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\text {Timed out} \] Input:
integrate(1/((b**2+c**2)**(1/2)+b*cos(e*x+d)+c*sin(e*x+d))**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="ma xima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}\right )}^{4}} \,d x } \] Input:
integrate(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="gi ac")
Output:
sage0*x
Time = 25.67 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.88 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx =\text {Too large to display} \] Input:
int(1/(b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^4,x)
Output:
-(tan(d/2 + (e*x)/2)^6*((16*b^4 + 2*c^4 + 16*b^2*c^2)/c^8 + ((8*b*c^2 + 16 *b^3)*(b^2 + c^2)^(1/2))/c^8) + tan(d/2 + (e*x)/2)*(((114*b*c^8)/5 + 512*b ^9 + (1352*b^3*c^6)/5 + (4576*b^5*c^4)/5 + (5888*b^7*c^2)/5)/c^13 + ((b^2 + c^2)^(1/2)*(512*b^8 + (14*c^8)/5 + (472*b^2*c^6)/5 + (2592*b^4*c^4)/5 + (4608*b^6*c^2)/5))/c^13) + ((1024*b^10)/7 + (24*c^10)/35 + (136*b^2*c^8)/7 + (4496*b^4*c^6)/35 + (11776*b^6*c^4)/35 + (13056*b^8*c^2)/35)/c^14 + tan (d/2 + (e*x)/2)^2*((768*b^8 + (42*c^8)/5 + (1116*b^2*c^6)/5 + (5136*b^4*c^ 4)/5 + (7872*b^6*c^2)/5)/c^12 + ((b^2 + c^2)^(1/2)*(60*b*c^6 + 768*b^7 + 5 28*b^3*c^4 + (5952*b^5*c^2)/5))/c^12) + tan(d/2 + (e*x)/2)^3*((80*b*c^6 + 640*b^7 + 600*b^3*c^4 + 1152*b^5*c^2)/c^11 + ((b^2 + c^2)^(1/2)*(640*b^6 + 12*c^6 + 264*b^2*c^4 + 832*b^4*c^2))/c^11) + tan(d/2 + (e*x)/2)^4*((320*b ^6 + 12*c^6 + 196*b^2*c^4 + 496*b^4*c^2)/c^10 + ((b^2 + c^2)^(1/2)*(68*b*c ^4 + 320*b^5 + 336*b^3*c^2))/c^10) + tan(d/2 + (e*x)/2)^5*((30*b*c^4 + 96* b^5 + 120*b^3*c^2)/c^9 + ((b^2 + c^2)^(1/2)*(96*b^4 + 6*c^4 + 72*b^2*c^2)) /c^9) + ((b^2 + c^2)^(1/2)*((24*b*c^8)/5 + (1024*b^9)/7 + (272*b^3*c^6)/5 + (1024*b^5*c^4)/5 + (10496*b^7*c^2)/35))/c^14)/(e*(tan(d/2 + (e*x)/2)^3*( (280*b^4 + 35*c^4 + 280*b^2*c^2)/c^4 + ((140*b*c^2 + 280*b^3)*(b^2 + c^2)^ (1/2))/c^4) + tan(d/2 + (e*x)/2)^4*((105*b*c^2 + 140*b^3)/c^3 + ((140*b^2 + 35*c^2)*(b^2 + c^2)^(1/2))/c^3) + tan(d/2 + (e*x)/2)^7 + tan(d/2 + (e*x) /2)*((224*b^6 + 7*c^6 + 126*b^2*c^4 + 336*b^4*c^2)/c^6 + ((b^2 + c^2)^(...
\[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4} \, dx=\int \frac {1}{\left (\sqrt {b^{2}+c^{2}}+b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{4}}d x \] Input:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x)
Output:
int(1/((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x)