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

Optimal. Leaf size=246 \[ \frac{35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac{35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}+\frac{35}{8} x \left (b^2+c^2\right )^2 \]

[Out]

(35*(b^2 + c^2)^2*x)/8 - (35*c*(b^2 + c^2)^(3/2)*Cos[d + e*x])/(8*e) + (35*b*(b^2 + c^2)^(3/2)*Sin[d + e*x])/(
8*e) - (35*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/
(24*e) - (7*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*
x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e
)

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Rubi [A]  time = 0.169314, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3113, 2637, 2638} \[ \frac{35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac{35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}+\frac{35}{8} x \left (b^2+c^2\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(b^2 + c^2)^2*x)/8 - (35*c*(b^2 + c^2)^(3/2)*Cos[d + e*x])/(8*e) + (35*b*(b^2 + c^2)^(3/2)*Sin[d + e*x])/(
8*e) - (35*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/
(24*e) - (7*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*
x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e
)

Rule 3113

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 - 1))/(e*n), x] + Dist[(a*(2*n - 1))/n, 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]
&& GtQ[n, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \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)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac{1}{4} \left (7 \sqrt{b^2+c^2}\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3 \, dx\\ &=-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac{1}{12} \left (35 \left (b^2+c^2\right )\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2 \, dx\\ &=-\frac{35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac{1}{8} \left (35 \left (b^2+c^2\right )^{3/2}\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right ) \, dx\\ &=\frac{35}{8} \left (b^2+c^2\right )^2 x-\frac{35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac{1}{8} \left (35 b \left (b^2+c^2\right )^{3/2}\right ) \int \cos (d+e x) \, dx+\frac{1}{8} \left (35 c \left (b^2+c^2\right )^{3/2}\right ) \int \sin (d+e x) \, dx\\ &=\frac{35}{8} \left (b^2+c^2\right )^2 x-\frac{35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}+\frac{35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac{35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\\ \end{align*}

Mathematica [C]  time = 1.48369, size = 238, normalized size = 0.97 \[ \frac{420 \left (b^2+c^2\right )^2 (d+e x)+672 b (b-i c) (b+i c) \sqrt{b^2+c^2} \sin (d+e x)+32 b \left (b^2-3 c^2\right ) \sqrt{b^2+c^2} \sin (3 (d+e x))+168 \left (b^4-c^4\right ) \sin (2 (d+e x))+3 \left (-6 b^2 c^2+b^4+c^4\right ) \sin (4 (d+e x))-336 b c \left (b^2+c^2\right ) \cos (2 (d+e x))-672 c (b-i c) (b+i c) \sqrt{b^2+c^2} \cos (d+e x)+32 c \left (c^2-3 b^2\right ) \sqrt{b^2+c^2} \cos (3 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))}{96 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420*(b^2 + c^2)^2*(d + e*x) - 672*(b - I*c)*(b + I*c)*c*Sqrt[b^2 + c^2]*Cos[d + e*x] - 336*b*c*(b^2 + c^2)*Co
s[2*(d + e*x)] + 32*c*(-3*b^2 + c^2)*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4*(d + e*x)] +
672*b*(b - I*c)*(b + I*c)*Sqrt[b^2 + c^2]*Sin[d + e*x] + 168*(b^4 - c^4)*Sin[2*(d + e*x)] + 32*b*(b^2 - 3*c^2)
*Sqrt[b^2 + c^2]*Sin[3*(d + e*x)] + 3*(b^4 - 6*b^2*c^2 + c^4)*Sin[4*(d + e*x)])/(96*e)

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Maple [B]  time = 0.124, size = 514, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(2*b^2*c^2*(e*x+d)+c^4*(-1/4*(sin(e*x+d)^3+3/2*sin(e*x+d))*cos(e*x+d)+3/8*e*x+3/8*d)+6*c^4*(-1/2*sin(e*x+d
)*cos(e*x+d)+1/2*e*x+1/2*d)+b^4*(1/4*(cos(e*x+d)^3+3/2*cos(e*x+d))*sin(e*x+d)+3/8*e*x+3/8*d)+6*b^4*(1/2*sin(e*
x+d)*cos(e*x+d)+1/2*e*x+1/2*d)+b^4*(e*x+d)+c^4*(e*x+d)+4/3*(b^2+c^2)^(1/2)*b^3*(2+cos(e*x+d)^2)*sin(e*x+d)+6*b
^2*c^2*(1/2*sin(e*x+d)*cos(e*x+d)+1/2*e*x+1/2*d)+4*(b^2+c^2)^(1/2)*b^3*sin(e*x+d)-4/3*(b^2+c^2)^(1/2)*c^3*(2+s
in(e*x+d)^2)*cos(e*x+d)+6*b^2*c^2*(-1/2*sin(e*x+d)*cos(e*x+d)+1/2*e*x+1/2*d)-4*(b^2+c^2)^(1/2)*c^3*cos(e*x+d)-
4*(b^2+c^2)^(1/2)*b^2*c*cos(e*x+d)^3+4*(b^2+c^2)^(1/2)*b*c^2*sin(e*x+d)^3-cos(e*x+d)^4*b^3*c+6*b^2*c^2*(-1/4*s
in(e*x+d)*cos(e*x+d)^3+1/8*sin(e*x+d)*cos(e*x+d)+1/8*e*x+1/8*d)+b*c^3*sin(e*x+d)^4-6*cos(e*x+d)^2*b^3*c-6*cos(
e*x+d)^2*b*c^3+4*(b^2+c^2)^(1/2)*b*c^2*sin(e*x+d)-4*(b^2+c^2)^(1/2)*b^2*c*cos(e*x+d))

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Maxima [A]  time = 1.01463, size = 478, normalized size = 1.94 \begin{align*} -\frac{b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac{b c^{3} \sin \left (e x + d\right )^{4}}{e} + \frac{{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac{3 \,{\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac{{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} +{\left (b^{2} + c^{2}\right )}^{2} x - 4 \,{\left (b^{2} + c^{2}\right )}^{\frac{3}{2}}{\left (\frac{c \cos \left (e x + d\right )}{e} - \frac{b \sin \left (e x + d\right )}{e}\right )} - \frac{3}{2} \,{\left (\frac{4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac{{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac{{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )}{\left (b^{2} + c^{2}\right )} - \frac{4}{3} \,{\left (\frac{3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac{3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac{{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac{{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} \sqrt{b^{2} + c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*c*cos(e*x + d)^4/e + b*c^3*sin(e*x + d)^4/e + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) + 8*sin(2*e*x + 2*d)
)*b^4/e + 3/16*(4*e*x + 4*d - sin(4*e*x + 4*d))*b^2*c^2/e + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) - 8*sin(2*e
*x + 2*d))*c^4/e + (b^2 + c^2)^2*x - 4*(b^2 + c^2)^(3/2)*(c*cos(e*x + d)/e - b*sin(e*x + d)/e) - 3/2*(4*b*c*co
s(e*x + d)^2/e - (2*e*x + 2*d + sin(2*e*x + 2*d))*b^2/e - (2*e*x + 2*d - sin(2*e*x + 2*d))*c^2/e)*(b^2 + c^2)
- 4/3*(3*b^2*c*cos(e*x + d)^3/e - 3*b*c^2*sin(e*x + d)^3/e + (sin(e*x + d)^3 - 3*sin(e*x + d))*b^3/e - (cos(e*
x + d)^3 - 3*cos(e*x + d))*c^3/e)*sqrt(b^2 + c^2)

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Fricas [A]  time = 2.32549, size = 509, normalized size = 2.07 \begin{align*} -\frac{24 \,{\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} - 105 \,{\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} e x + 48 \,{\left (3 \, b^{3} c + 4 \, b c^{3}\right )} \cos \left (e x + d\right )^{2} - 3 \,{\left (2 \,{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} +{\left (27 \, b^{4} + 6 \, b^{2} c^{2} - 29 \, c^{4}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 32 \,{\left ({\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + 3 \,{\left (b^{2} c + 2 \, c^{3}\right )} \cos \left (e x + d\right ) -{\left (5 \, b^{3} + 6 \, b c^{2} +{\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt{b^{2} + c^{2}}}{24 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(24*(b^3*c - b*c^3)*cos(e*x + d)^4 - 105*(b^4 + 2*b^2*c^2 + c^4)*e*x + 48*(3*b^3*c + 4*b*c^3)*cos(e*x +
d)^2 - 3*(2*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^3 + (27*b^4 + 6*b^2*c^2 - 29*c^4)*cos(e*x + d))*sin(e*x + d)
+ 32*((3*b^2*c - c^3)*cos(e*x + d)^3 + 3*(b^2*c + 2*c^3)*cos(e*x + d) - (5*b^3 + 6*b*c^2 + (b^3 - 3*b*c^2)*cos
(e*x + d)^2)*sin(e*x + d))*sqrt(b^2 + c^2))/e

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Sympy [A]  time = 4.22427, size = 882, normalized size = 3.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*b**4*x*sin(d + e*x)**4/8 + 3*b**4*x*sin(d + e*x)**2*cos(d + e*x)**2/4 + 3*b**4*x*sin(d + e*x)**2
+ 3*b**4*x*cos(d + e*x)**4/8 + 3*b**4*x*cos(d + e*x)**2 + b**4*x + 3*b**4*sin(d + e*x)**3*cos(d + e*x)/(8*e) +
5*b**4*sin(d + e*x)*cos(d + e*x)**3/(8*e) + 3*b**4*sin(d + e*x)*cos(d + e*x)/e + 6*b**3*c*sin(d + e*x)**2/e -
b**3*c*cos(d + e*x)**4/e + 8*b**3*sqrt(b**2 + c**2)*sin(d + e*x)**3/(3*e) + 4*b**3*sqrt(b**2 + c**2)*sin(d +
e*x)*cos(d + e*x)**2/e + 4*b**3*sqrt(b**2 + c**2)*sin(d + e*x)/e + 3*b**2*c**2*x*sin(d + e*x)**4/4 + 3*b**2*c*
*2*x*sin(d + e*x)**2*cos(d + e*x)**2/2 + 6*b**2*c**2*x*sin(d + e*x)**2 + 3*b**2*c**2*x*cos(d + e*x)**4/4 + 6*b
**2*c**2*x*cos(d + e*x)**2 + 2*b**2*c**2*x + 3*b**2*c**2*sin(d + e*x)**3*cos(d + e*x)/(4*e) - 3*b**2*c**2*sin(
d + e*x)*cos(d + e*x)**3/(4*e) - 4*b**2*c*sqrt(b**2 + c**2)*cos(d + e*x)**3/e - 4*b**2*c*sqrt(b**2 + c**2)*cos
(d + e*x)/e - 2*b*c**3*sin(d + e*x)**2*cos(d + e*x)**2/e + 6*b*c**3*sin(d + e*x)**2/e - b*c**3*cos(d + e*x)**4
/e + 4*b*c**2*sqrt(b**2 + c**2)*sin(d + e*x)**3/e + 4*b*c**2*sqrt(b**2 + c**2)*sin(d + e*x)/e + 3*c**4*x*sin(d
+ e*x)**4/8 + 3*c**4*x*sin(d + e*x)**2*cos(d + e*x)**2/4 + 3*c**4*x*sin(d + e*x)**2 + 3*c**4*x*cos(d + e*x)**
4/8 + 3*c**4*x*cos(d + e*x)**2 + c**4*x - 5*c**4*sin(d + e*x)**3*cos(d + e*x)/(8*e) - 3*c**4*sin(d + e*x)*cos(
d + e*x)**3/(8*e) - 3*c**4*sin(d + e*x)*cos(d + e*x)/e - 4*c**3*sqrt(b**2 + c**2)*sin(d + e*x)**2*cos(d + e*x)
/e - 8*c**3*sqrt(b**2 + c**2)*cos(d + e*x)**3/(3*e) - 4*c**3*sqrt(b**2 + c**2)*cos(d + e*x)/e, Ne(e, 0)), (x*(
b*cos(d) + c*sin(d) + sqrt(b**2 + c**2))**4, True))

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Giac [A]  time = 1.24363, size = 387, normalized size = 1.57 \begin{align*} -\frac{1}{8} \,{\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, x e + 4 \, d\right ) e^{\left (-1\right )} - \frac{1}{3} \,{\left (3 \, \sqrt{b^{2} + c^{2}} b^{2} c - \sqrt{b^{2} + c^{2}} c^{3}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac{7}{2} \,{\left (b^{3} c + b c^{3}\right )} \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - 7 \,{\left (\sqrt{b^{2} + c^{2}} b^{2} c + \sqrt{b^{2} + c^{2}} c^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} + \frac{1}{32} \,{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} e^{\left (-1\right )} \sin \left (4 \, x e + 4 \, d\right ) + \frac{1}{3} \,{\left (\sqrt{b^{2} + c^{2}} b^{3} - 3 \, \sqrt{b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) + \frac{7}{4} \,{\left (b^{4} - c^{4}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 7 \,{\left (\sqrt{b^{2} + c^{2}} b^{3} + \sqrt{b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) + \frac{35}{8} \,{\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b^3*c - b*c^3)*cos(4*x*e + 4*d)*e^(-1) - 1/3*(3*sqrt(b^2 + c^2)*b^2*c - sqrt(b^2 + c^2)*c^3)*cos(3*x*e +
3*d)*e^(-1) - 7/2*(b^3*c + b*c^3)*cos(2*x*e + 2*d)*e^(-1) - 7*(sqrt(b^2 + c^2)*b^2*c + sqrt(b^2 + c^2)*c^3)*c
os(x*e + d)*e^(-1) + 1/32*(b^4 - 6*b^2*c^2 + c^4)*e^(-1)*sin(4*x*e + 4*d) + 1/3*(sqrt(b^2 + c^2)*b^3 - 3*sqrt(
b^2 + c^2)*b*c^2)*e^(-1)*sin(3*x*e + 3*d) + 7/4*(b^4 - c^4)*e^(-1)*sin(2*x*e + 2*d) + 7*(sqrt(b^2 + c^2)*b^3 +
sqrt(b^2 + c^2)*b*c^2)*e^(-1)*sin(x*e + d) + 35/8*(b^4 + 2*b^2*c^2 + c^4)*x