3.362 \(\int \frac {1}{(\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x))^4} \, dx\)

Optimal. Leaf size=259 \[ -\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right )^{3/2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right ) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}+\frac {b \sin (d+e x)-c \cos (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{35 c e \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))} \]

[Out]

1/7*(-c*cos(e*x+d)+b*sin(e*x+d))/e/(b^2+c^2)^(1/2)/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4-3/35*(c*cos(e
*x+d)-b*sin(e*x+d))/(b^2+c^2)/e/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^3-2/35*(c*cos(e*x+d)-b*sin(e*x+d))
/(b^2+c^2)^(3/2)/e/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^2-2/35*(c-sin(e*x+d)*(b^2+c^2)^(1/2))/c/(b^2+c^
2)^(3/2)/e/(c*cos(e*x+d)-b*sin(e*x+d))

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Rubi [A]  time = 0.19, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3116, 3114} \[ -\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right )^{3/2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left (b^2+c^2\right ) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}-\frac {c \cos (d+e x)-b \sin (d+e x)}{7 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4}-\frac {2 \left (c-\sqrt {b^2+c^2} \sin (d+e x)\right )}{35 c e \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

[Out]

-(c*Cos[d + e*x] - b*Sin[d + e*x])/(7*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4)
- (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])
^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)^(3/2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[
d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(35*c*(b^2 + c^2)^(3/2)*e*(c*Cos[d + e*x] - b*Sin[d + e*
x]))

Rule 3114

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]

Rule 3116

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] + Dist[(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]

Rubi steps

\begin {align*} \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 \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3} \, dx}{7 \sqrt {b^2+c^2}}\\ &=-\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 {6 \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2} \, dx}{35 \left (b^2+c^2\right )}\\ &=-\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 \int \frac {1}{\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx}{35 \left (b^2+c^2\right )^{3/2}}\\ &=-\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))}\\ \end {align*}

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Mathematica [B]  time = 2.03, size = 533, normalized size = 2.06 \[ \frac {-35 b^6 \sin (d+e x)+21 b^6 \sin (3 (d+e x))-7 b^6 \sin (5 (d+e x))+b^6 \sin (7 (d+e x))-112 b^5 c \cos (3 (d+e x))+28 b^5 c \cos (5 (d+e x))-6 b^5 c \cos (7 (d+e x))-1295 b^4 c^2 \sin (d+e x)-189 b^4 c^2 \sin (3 (d+e x))+35 b^4 c^2 \sin (5 (d+e x))-15 b^4 c^2 \sin (7 (d+e x))+56 b^3 c^3 \cos (3 (d+e x))+20 b^3 c^3 \cos (7 (d+e x))-2485 b^2 c^4 \sin (d+e x)-161 b^2 c^4 \sin (3 (d+e x))+35 b^2 c^4 \sin (5 (d+e x))+15 b^2 c^4 \sin (7 (d+e x))-1190 b c \left (b^2+c^2\right )^2 \cos (d+e x)+832 c^5 \sqrt {b^2+c^2}+896 b c^4 \sqrt {b^2+c^2} \sin (2 (d+e x))+1664 b^2 c^3 \sqrt {b^2+c^2}+832 b^4 c \sqrt {b^2+c^2}+448 c \sqrt {b^2+c^2} \left (b^4-c^4\right ) \cos (2 (d+e x))+896 b^3 c^2 \sqrt {b^2+c^2} \sin (2 (d+e x))+168 b c^5 \cos (3 (d+e x))-28 b c^5 \cos (5 (d+e x))-6 b c^5 \cos (7 (d+e x))-1225 c^6 \sin (d+e x)+49 c^6 \sin (3 (d+e x))-7 c^6 \sin (5 (d+e x))-c^6 \sin (7 (d+e x))}{1120 c e \left (b^2+c^2\right ) (b \sin (d+e x)-c \cos (d+e x))^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

[Out]

(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*C
os[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*C
os[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*(b^2 + c^2)*e*(-(c*Cos[d + e*x]) + b*Sin[d + e*x])^7)

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fricas [B]  time = 1.85, size = 739, normalized size = 2.85 \[ \frac {2 \, {\left (b^{7} - 21 \, b^{5} c^{2} + 35 \, b^{3} c^{4} - 7 \, b c^{6}\right )} \cos \left (e x + d\right )^{7} - 7 \, {\left (b^{7} - 15 \, b^{5} c^{2} + 15 \, b^{3} c^{4} - b c^{6}\right )} \cos \left (e x + d\right )^{5} - 14 \, {\left (5 \, b^{5} c^{2} - 5 \, b^{3} c^{4} - 2 \, b c^{6}\right )} \cos \left (e x + d\right )^{3} - 7 \, {\left (5 \, b^{7} + 15 \, b^{5} c^{2} + 20 \, b^{3} c^{4} + 8 \, b c^{6}\right )} \cos \left (e x + d\right ) - {\left (35 \, b^{6} c + 105 \, b^{4} c^{3} + 112 \, b^{2} c^{5} + 40 \, c^{7} - 2 \, {\left (7 \, b^{6} c - 35 \, b^{4} c^{3} + 21 \, b^{2} c^{5} - c^{7}\right )} \cos \left (e x + d\right )^{6} + {\left (35 \, b^{6} c - 105 \, b^{4} c^{3} + 21 \, b^{2} c^{5} + c^{7}\right )} \cos \left (e x + d\right )^{4} + 2 \, {\left (35 \, b^{4} c^{3} + 7 \, b^{2} c^{5} - 4 \, c^{7}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) + 4 \, {\left (3 \, b^{6} + 16 \, b^{4} c^{2} + 23 \, b^{2} c^{4} + 10 \, c^{6} + 7 \, {\left (b^{6} + b^{4} c^{2} - b^{2} c^{4} - c^{6}\right )} \cos \left (e x + d\right )^{2} + 14 \, {\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 )\right )} \sqrt {b^{2} + c^{2}}}{35 \, {\left ({\left (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}\right )} e \cos \left (e x + d\right )^{7} - 7 \, {\left (3 \, b^{10} c - 4 \, b^{8} c^{3} - 14 \, b^{6} c^{5} - 4 \, b^{4} c^{7} + 3 \, b^{2} c^{9}\right )} e \cos \left (e x + d\right )^{5} + 7 \, {\left (3 \, b^{10} c + b^{8} c^{3} - 7 \, b^{6} c^{5} - 5 \, b^{4} c^{7}\right )} e \cos \left (e x + d\right )^{3} - 7 \, {\left (b^{10} c + 2 \, b^{8} c^{3} + b^{6} c^{5}\right )} e \cos \left (e x + d\right ) - {\left ({\left (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}\right )} e \cos \left (e x + d\right )^{6} - {\left (3 \, b^{11} - 36 \, b^{9} c^{2} - 46 \, b^{7} c^{4} + 28 \, b^{5} c^{6} + 35 \, b^{3} c^{8}\right )} e \cos \left (e x + d\right )^{4} + 3 \, {\left (b^{11} - 5 \, b^{9} c^{2} - 13 \, b^{7} c^{4} - 7 \, b^{5} c^{6}\right )} e \cos \left (e x + d\right )^{2} - {\left (b^{11} + 2 \, b^{9} c^{2} + b^{7} c^{4}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="fricas")

[Out]

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)*c
os(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 - 36*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))

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giac [B]  time = 2.52, size = 599, normalized size = 2.31 \[ -\frac {2 \, {\left (2560 \, b^{10} + 6528 \, b^{8} c^{2} + 5888 \, b^{6} c^{4} + 2248 \, b^{4} c^{6} + 340 \, b^{2} c^{8} + 12 \, c^{10} + 35 \, {\left (8 \, b^{4} c^{6} + 8 \, b^{2} c^{8} + c^{10} + 4 \, {\left (2 \, b^{3} c^{6} + b c^{8}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{6} + 105 \, {\left (16 \, b^{5} c^{5} + 20 \, b^{3} c^{7} + 5 \, b c^{9} + {\left (16 \, b^{4} c^{5} + 12 \, b^{2} c^{7} + c^{9}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 70 \, {\left (80 \, b^{6} c^{4} + 124 \, b^{4} c^{6} + 49 \, b^{2} c^{8} + 3 \, c^{10} + {\left (80 \, b^{5} c^{4} + 84 \, b^{3} c^{6} + 17 \, b c^{8}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 70 \, {\left (160 \, b^{7} c^{3} + 288 \, b^{5} c^{5} + 150 \, b^{3} c^{7} + 20 \, b c^{9} + {\left (160 \, b^{6} c^{3} + 208 \, b^{4} c^{5} + 66 \, b^{2} c^{7} + 3 \, c^{9}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 21 \, {\left (640 \, b^{8} c^{2} + 1312 \, b^{6} c^{4} + 856 \, b^{4} c^{6} + 186 \, b^{2} c^{8} + 7 \, c^{10} + 2 \, {\left (320 \, b^{7} c^{2} + 496 \, b^{5} c^{4} + 220 \, b^{3} c^{6} + 25 \, b c^{8}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 7 \, {\left (1280 \, b^{9} c + 2944 \, b^{7} c^{3} + 2288 \, b^{5} c^{5} + 676 \, b^{3} c^{7} + 57 \, b c^{9} + {\left (1280 \, b^{8} c + 2304 \, b^{6} c^{3} + 1296 \, b^{4} c^{5} + 236 \, b^{2} c^{7} + 7 \, c^{9}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 4 \, {\left (640 \, b^{9} + 1312 \, b^{7} c^{2} + 896 \, b^{5} c^{4} + 238 \, b^{3} c^{6} + 21 \, b c^{8}\right )} \sqrt {b^{2} + c^{2}}\right )} e^{\left (-1\right )}}{35 \, {\left (c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b + \sqrt {b^{2} + c^{2}}\right )}^{7} c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="giac")

[Out]

-2/35*(2560*b^10 + 6528*b^8*c^2 + 5888*b^6*c^4 + 2248*b^4*c^6 + 340*b^2*c^8 + 12*c^10 + 35*(8*b^4*c^6 + 8*b^2*
c^8 + c^10 + 4*(2*b^3*c^6 + b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^6 + 105*(16*b^5*c^5 + 20*b^3*c^7 + 5*
b*c^9 + (16*b^4*c^5 + 12*b^2*c^7 + c^9)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^5 + 70*(80*b^6*c^4 + 124*b^4*c^6
+ 49*b^2*c^8 + 3*c^10 + (80*b^5*c^4 + 84*b^3*c^6 + 17*b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^4 + 70*(16
0*b^7*c^3 + 288*b^5*c^5 + 150*b^3*c^7 + 20*b*c^9 + (160*b^6*c^3 + 208*b^4*c^5 + 66*b^2*c^7 + 3*c^9)*sqrt(b^2 +
c^2))*tan(1/2*x*e + 1/2*d)^3 + 21*(640*b^8*c^2 + 1312*b^6*c^4 + 856*b^4*c^6 + 186*b^2*c^8 + 7*c^10 + 2*(320*b
^7*c^2 + 496*b^5*c^4 + 220*b^3*c^6 + 25*b*c^8)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d)^2 + 7*(1280*b^9*c + 2944*
b^7*c^3 + 2288*b^5*c^5 + 676*b^3*c^7 + 57*b*c^9 + (1280*b^8*c + 2304*b^6*c^3 + 1296*b^4*c^5 + 236*b^2*c^7 + 7*
c^9)*sqrt(b^2 + c^2))*tan(1/2*x*e + 1/2*d) + 4*(640*b^9 + 1312*b^7*c^2 + 896*b^5*c^4 + 238*b^3*c^6 + 21*b*c^8)
*sqrt(b^2 + c^2))*e^(-1)/((c*tan(1/2*x*e + 1/2*d) + b + sqrt(b^2 + c^2))^7*c^7)

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maple [B]  time = 0.71, size = 823, normalized size = 3.18 \[ -\frac {2 \left (\frac {\left (8 b^{4}+8 b^{2} c^{2}+c^{4}+8 \sqrt {b^{2}+c^{2}}\, b^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2}\right ) \left (\tan ^{6}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{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 c^{4} b \right ) \left (\tan ^{5}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{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 b^{4} c^{2}+49 b^{2} c^{4}+3 c^{6}\right ) \left (\tan ^{4}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{c^{4}}+\frac {2 \left (160 b^{7}+288 b^{5} c^{2}+150 b^{3} c^{4}+20 c^{6} b +160 \sqrt {b^{2}+c^{2}}\, b^{6}+208 \sqrt {b^{2}+c^{2}}\, b^{4} c^{2}+66 \sqrt {b^{2}+c^{2}}\, b^{2} c^{4}+3 \sqrt {b^{2}+c^{2}}\, c^{6}\right ) \left (\tan ^{3}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{c^{5}}+\frac {3 \left (640 b^{7} \sqrt {b^{2}+c^{2}}+992 \sqrt {b^{2}+c^{2}}\, b^{5} c^{2}+440 \sqrt {b^{2}+c^{2}}\, b^{3} c^{4}+50 \sqrt {b^{2}+c^{2}}\, b \,c^{6}+640 b^{8}+1312 b^{6} c^{2}+856 b^{4} c^{4}+186 b^{2} c^{6}+7 c^{8}\right ) \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{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 {d}{2}+\frac {e x}{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 ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{c}+\frac {2 b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{c}+\frac {2 \sqrt {b^{2}+c^{2}}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right )^{3} \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x)

[Out]

-2/e/c^6*((8*b^4+8*b^2*c^2+c^4+8*(b^2+c^2)^(1/2)*b^3+4*(b^2+c^2)^(1/2)*b*c^2)/c^2*tan(1/2*d+1/2*e*x)^6+3*(16*(
b^2+c^2)^(1/2)*b^4+12*(b^2+c^2)^(1/2)*b^2*c^2+(b^2+c^2)^(1/2)*c^4+16*b^5+20*b^3*c^2+5*c^4*b)/c^3*tan(1/2*d+1/2
*e*x)^5+2*(80*(b^2+c^2)^(1/2)*b^5+84*(b^2+c^2)^(1/2)*b^3*c^2+17*(b^2+c^2)^(1/2)*b*c^4+80*b^6+124*b^4*c^2+49*b^
2*c^4+3*c^6)/c^4*tan(1/2*d+1/2*e*x)^4+2*(160*b^7+288*b^5*c^2+150*b^3*c^4+20*c^6*b+160*(b^2+c^2)^(1/2)*b^6+208*
(b^2+c^2)^(1/2)*b^4*c^2+66*(b^2+c^2)^(1/2)*b^2*c^4+3*(b^2+c^2)^(1/2)*c^6)/c^5*tan(1/2*d+1/2*e*x)^3+3/5*(640*b^
7*(b^2+c^2)^(1/2)+992*(b^2+c^2)^(1/2)*b^5*c^2+440*(b^2+c^2)^(1/2)*b^3*c^4+50*(b^2+c^2)^(1/2)*b*c^6+640*b^8+131
2*b^6*c^2+856*b^4*c^4+186*b^2*c^6+7*c^8)/c^6*tan(1/2*d+1/2*e*x)^2+1/5*(1280*b^9+2944*b^7*c^2+2288*b^5*c^4+676*
b^3*c^6+57*b*c^8+1280*(b^2+c^2)^(1/2)*b^8+2304*(b^2+c^2)^(1/2)*b^6*c^2+1296*(b^2+c^2)^(1/2)*b^4*c^4+236*(b^2+c
^2)^(1/2)*b^2*c^6+7*(b^2+c^2)^(1/2)*c^8)/c^7*tan(1/2*d+1/2*e*x)+4/35*(640*(b^2+c^2)^(1/2)*b^9+1312*(b^2+c^2)^(
1/2)*b^7*c^2+896*(b^2+c^2)^(1/2)*b^5*c^4+238*(b^2+c^2)^(1/2)*b^3*c^6+21*(b^2+c^2)^(1/2)*b*c^8+640*b^10+1632*b^
8*c^2+1472*b^6*c^4+562*b^4*c^6+85*b^2*c^8+3*c^10)/c^8)/(tan(1/2*d+1/2*e*x)^2+2/c*(b^2+c^2)^(1/2)*tan(1/2*d+1/2
*e*x)+2*b/c*tan(1/2*d+1/2*e*x)+2/c^2*(b^2+c^2)^(1/2)*b+2/c^2*b^2+1)^3/(tan(1/2*d+1/2*e*x)+1/c*(b^2+c^2)^(1/2)+
b/c)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 12.31, size = 1004, normalized size = 3.88 \[ -\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (\frac {16\,b^4+16\,b^2\,c^2+2\,c^4}{c^8}+\frac {\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^8}\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {512\,b^9+\frac {5888\,b^7\,c^2}{5}+\frac {4576\,b^5\,c^4}{5}+\frac {1352\,b^3\,c^6}{5}+\frac {114\,b\,c^8}{5}}{c^{13}}+\frac {\sqrt {b^2+c^2}\,\left (512\,b^8+\frac {4608\,b^6\,c^2}{5}+\frac {2592\,b^4\,c^4}{5}+\frac {472\,b^2\,c^6}{5}+\frac {14\,c^8}{5}\right )}{c^{13}}\right )+\frac {\frac {1024\,b^{10}}{7}+\frac {13056\,b^8\,c^2}{35}+\frac {11776\,b^6\,c^4}{35}+\frac {4496\,b^4\,c^6}{35}+\frac {136\,b^2\,c^8}{7}+\frac {24\,c^{10}}{35}}{c^{14}}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {768\,b^8+\frac {7872\,b^6\,c^2}{5}+\frac {5136\,b^4\,c^4}{5}+\frac {1116\,b^2\,c^6}{5}+\frac {42\,c^8}{5}}{c^{12}}+\frac {\sqrt {b^2+c^2}\,\left (768\,b^7+\frac {5952\,b^5\,c^2}{5}+528\,b^3\,c^4+60\,b\,c^6\right )}{c^{12}}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {640\,b^7+1152\,b^5\,c^2+600\,b^3\,c^4+80\,b\,c^6}{c^{11}}+\frac {\sqrt {b^2+c^2}\,\left (640\,b^6+832\,b^4\,c^2+264\,b^2\,c^4+12\,c^6\right )}{c^{11}}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (\frac {320\,b^6+496\,b^4\,c^2+196\,b^2\,c^4+12\,c^6}{c^{10}}+\frac {\sqrt {b^2+c^2}\,\left (320\,b^5+336\,b^3\,c^2+68\,b\,c^4\right )}{c^{10}}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (\frac {96\,b^5+120\,b^3\,c^2+30\,b\,c^4}{c^9}+\frac {\sqrt {b^2+c^2}\,\left (96\,b^4+72\,b^2\,c^2+6\,c^4\right )}{c^9}\right )+\frac {\sqrt {b^2+c^2}\,\left (\frac {1024\,b^9}{7}+\frac {10496\,b^7\,c^2}{35}+\frac {1024\,b^5\,c^4}{5}+\frac {272\,b^3\,c^6}{5}+\frac {24\,b\,c^8}{5}\right )}{c^{14}}}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\frac {280\,b^4+280\,b^2\,c^2+35\,c^4}{c^4}+\frac {\left (280\,b^3+140\,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 {140\,b^3+105\,b\,c^2}{c^3}+\frac {\left (140\,b^2+35\,c^2\right )\,\sqrt {b^2+c^2}}{c^3}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^7+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {224\,b^6+336\,b^4\,c^2+126\,b^2\,c^4+7\,c^6}{c^6}+\frac {\sqrt {b^2+c^2}\,\left (224\,b^5+224\,b^3\,c^2+42\,b\,c^4\right )}{c^6}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (\frac {42\,b^2+21\,c^2}{c^2}+\frac {42\,b\,\sqrt {b^2+c^2}}{c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (\frac {7\,\sqrt {b^2+c^2}}{c}+\frac {7\,b}{c}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {336\,b^5+420\,b^3\,c^2+105\,b\,c^4}{c^5}+\frac {\sqrt {b^2+c^2}\,\left (336\,b^4+252\,b^2\,c^2+21\,c^4\right )}{c^5}\right )+\frac {64\,b^7+112\,b^5\,c^2+56\,b^3\,c^4+7\,b\,c^6}{c^7}+\frac {\sqrt {b^2+c^2}\,\left (64\,b^6+80\,b^4\,c^2+24\,b^2\,c^4+c^6\right )}{c^7}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^4,x)

[Out]

-(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 + 528*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)^(1/2)*(42*b*c^4 + 224*b^5 + 224*b^3*c^2))/c^6) + tan(d/2 + (e*x)
/2)^5*((42*b^2 + 21*c^2)/c^2 + (42*b*(b^2 + c^2)^(1/2))/c^2) + tan(d/2 + (e*x)/2)^6*((7*(b^2 + c^2)^(1/2))/c +
(7*b)/c) + tan(d/2 + (e*x)/2)^2*((105*b*c^4 + 336*b^5 + 420*b^3*c^2)/c^5 + ((b^2 + c^2)^(1/2)*(336*b^4 + 21*c
^4 + 252*b^2*c^2))/c^5) + (7*b*c^6 + 64*b^7 + 56*b^3*c^4 + 112*b^5*c^2)/c^7 + ((b^2 + c^2)^(1/2)*(64*b^6 + c^6
+ 24*b^2*c^4 + 80*b^4*c^2))/c^7))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b**2+c**2)**(1/2))**4,x)

[Out]

Timed out

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