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]

35/8*(b^2+c^2)^2*x-35/8*c*(b^2+c^2)^(3/2)*cos(e*x+d)/e+35/8*b*(b^2+c^2)^(3/2)*sin(e*x+d)/e-35/24*(b^2+c^2)*(c*
cos(e*x+d)-b*sin(e*x+d))*(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))/e-7/12*(c*cos(e*x+d)-b*sin(e*x+d))*(b^2+c
^2)^(1/2)*(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^2/e-1/4*(c*cos(e*x+d)-b*sin(e*x+d))*(b*cos(e*x+d)+c*sin(
e*x+d)+(b^2+c^2)^(1/2))^3/e

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Rubi [A]  time = 0.17, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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]

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]

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*}

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Mathematica [C]  time = 1.42, size = 238, normalized size = 0.97 \[ \frac {168 \left (b^4-c^4\right ) \sin (2 (d+e x))+420 \left (b^2+c^2\right )^2 (d+e x)+672 b (b-i c) (b+i c) \sqrt {b^2+c^2} \sin (d+e x)+32 b \left (b^2-3 c^2\right ) \sqrt {b^2+c^2} \sin (3 (d+e x))-336 b c \left (b^2+c^2\right ) \cos (2 (d+e x))-672 c (b-i c) (b+i c) \sqrt {b^2+c^2} \cos (d+e x)+32 c \left (c^2-3 b^2\right ) \sqrt {b^2+c^2} \cos (3 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))+3 \left (b^4-6 b^2 c^2+c^4\right ) \sin (4 (d+e x))}{96 e} \]

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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fricas [A]  time = 1.06, size = 221, normalized size = 0.90 \[ -\frac {24 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} - 105 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} e x + 48 \, {\left (3 \, b^{3} c + 4 \, b c^{3}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (2 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} + {\left (27 \, b^{4} + 6 \, b^{2} c^{2} - 29 \, c^{4}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 32 \, {\left ({\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (b^{2} c + 2 \, c^{3}\right )} \cos \left (e x + d\right ) - {\left (5 \, b^{3} + 6 \, b c^{2} + {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}}{24 \, e} \]

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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giac [A]  time = 0.35, size = 287, normalized size = 1.17 \[ -\frac {1}{8} \, {\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, x e + 4 \, d\right ) e^{\left (-1\right )} - \frac {1}{3} \, {\left (3 \, \sqrt {b^{2} + c^{2}} b^{2} c - \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac {7}{2} \, {\left (b^{3} c + b c^{3}\right )} \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - 7 \, {\left (\sqrt {b^{2} + c^{2}} b^{2} c + \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} + \frac {1}{32} \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} e^{\left (-1\right )} \sin \left (4 \, x e + 4 \, d\right ) + \frac {1}{3} \, {\left (\sqrt {b^{2} + c^{2}} b^{3} - 3 \, \sqrt {b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) + \frac {7}{4} \, {\left (b^{4} - c^{4}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 7 \, {\left (\sqrt {b^{2} + c^{2}} b^{3} + \sqrt {b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) + \frac {35}{8} \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} x \]

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

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maple [B]  time = 0.30, size = 514, normalized size = 2.09 \[ \frac {b^{4} \left (e x +d \right )+2 b^{2} c^{2} \left (e x +d \right )+c^{4} \left (e x +d \right )+b^{4} \left (\frac {\left (\cos ^{3}\left (e x +d \right )+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+6 b^{4} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+c^{4} \left (-\frac {\left (\sin ^{3}\left (e x +d \right )+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+6 c^{4} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \left (\cos ^{3}\left (e x +d \right )\right )+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \left (\sin ^{3}\left (e x +d \right )\right )+\frac {4 \sqrt {b^{2}+c^{2}}\, b^{3} \left (2+\cos ^{2}\left (e x +d \right )\right ) \sin \left (e x +d \right )}{3}+6 b^{2} c^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+4 \sqrt {b^{2}+c^{2}}\, b^{3} \sin \left (e x +d \right )-\frac {4 \sqrt {b^{2}+c^{2}}\, c^{3} \left (2+\sin ^{2}\left (e x +d \right )\right ) \cos \left (e x +d \right )}{3}+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 \sqrt {b^{2}+c^{2}}\, c^{3} \cos \left (e x +d \right )-\left (\cos ^{4}\left (e x +d \right )\right ) b^{3} c +6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \left (\cos ^{3}\left (e x +d \right )\right )}{4}+\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \left (\sin ^{4}\left (e x +d \right )\right )-6 \left (\cos ^{2}\left (e x +d \right )\right ) b^{3} c -6 \left (\cos ^{2}\left (e x +d \right )\right ) b \,c^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )}{e} \]

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*(b^4*(e*x+d)+2*b^2*c^2*(e*x+d)+c^4*(e*x+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)+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)-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+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+sin(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)-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)+c^3*b*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 = 0.33, size = 354, normalized size = 1.44 \[ -\frac {b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac {b c^{3} \sin \left (e x + d\right )^{4}}{e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac {3 \, {\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} + {\left (b^{2} + c^{2}\right )}^{2} x - 4 \, {\left (b^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} - \frac {3}{2} \, {\left (\frac {4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} {\left (b^{2} + c^{2}\right )} - \frac {4}{3} \, {\left (\frac {3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac {3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac {{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac {{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} \sqrt {b^{2} + c^{2}} \]

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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mupad [B]  time = 7.32, size = 522, normalized size = 2.12 \[ \frac {35\,\mathrm {atan}\left (\frac {35\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,\left (\frac {35\,b^4}{4}+\frac {35\,b^2\,c^2}{2}+\frac {35\,c^4}{4}\right )}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}-\frac {35\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )-\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {29\,b^4}{4}-\frac {27\,c^4}{4}-\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+8\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (64\,b\,c^3+48\,b^3\,c-\left (48\,b^2\,c+40\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+\frac {136\,c^3}{3}\right )\,\sqrt {b^2+c^2}\right )-\left (16\,b^2\,c+\frac {40\,c^3}{3}\right )\,\sqrt {b^2+c^2}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^7\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {29\,b^4}{4}+\frac {27\,c^4}{4}+\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {21\,b^4}{4}-\frac {35\,c^4}{4}+\frac {21\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {21\,b^4}{4}+\frac {35\,c^4}{4}-\frac {21\,b^2\,c^2}{2}\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35*atan((35*tan(d/2 + (e*x)/2)*(b^2 + c^2)^2)/(4*((35*b^4)/4 + (35*c^4)/4 + (35*b^2*c^2)/2)))*(b^2 + c^2)^2)/
(4*e) - (35*(atan(tan(d/2 + (e*x)/2)) - (e*x)/2)*(b^2 + c^2)^2)/(4*e) + (tan(d/2 + (e*x)/2)*((8*b*c^2 + 16*b^3
)*(b^2 + c^2)^(1/2) + (29*b^4)/4 - (27*c^4)/4 - (3*b^2*c^2)/2) + tan(d/2 + (e*x)/2)^6*(24*b*c^3 + 32*b^3*c - (
32*b^2*c + 8*c^3)*(b^2 + c^2)^(1/2)) + tan(d/2 + (e*x)/2)^4*(64*b*c^3 + 48*b^3*c - (48*b^2*c + 40*c^3)*(b^2 +
c^2)^(1/2)) + tan(d/2 + (e*x)/2)^2*(24*b*c^3 + 32*b^3*c - (32*b^2*c + (136*c^3)/3)*(b^2 + c^2)^(1/2)) - (16*b^
2*c + (40*c^3)/3)*(b^2 + c^2)^(1/2) + tan(d/2 + (e*x)/2)^7*((8*b*c^2 + 16*b^3)*(b^2 + c^2)^(1/2) - (29*b^4)/4
+ (27*c^4)/4 + (3*b^2*c^2)/2) + tan(d/2 + (e*x)/2)^3*((56*b*c^2 + (112*b^3)/3)*(b^2 + c^2)^(1/2) + (21*b^4)/4
- (35*c^4)/4 + (21*b^2*c^2)/2) + tan(d/2 + (e*x)/2)^5*((56*b*c^2 + (112*b^3)/3)*(b^2 + c^2)^(1/2) - (21*b^4)/4
+ (35*c^4)/4 - (21*b^2*c^2)/2))/(e*(4*tan(d/2 + (e*x)/2)^2 + 6*tan(d/2 + (e*x)/2)^4 + 4*tan(d/2 + (e*x)/2)^6
+ tan(d/2 + (e*x)/2)^8 + 1))

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sympy [A]  time = 3.04, size = 857, normalized size = 3.48 \[ \begin {cases} \frac {3 b^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 b^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + 3 b^{4} x \sin ^{2}{\left (d + e x \right )} + \frac {3 b^{4} x \cos ^{4}{\left (d + e x \right )}}{8} + 3 b^{4} x \cos ^{2}{\left (d + e x \right )} + b^{4} x + \frac {3 b^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} + \frac {5 b^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} + \frac {3 b^{4} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {b^{3} c \cos ^{4}{\left (d + e x \right )}}{e} - \frac {6 b^{3} c \cos ^{2}{\left (d + e x \right )}}{e} + \frac {8 b^{3} \sqrt {b^{2} + c^{2}} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {4 b^{3} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {4 b^{3} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )}}{e} + \frac {3 b^{2} c^{2} x \sin ^{4}{\left (d + e x \right )}}{4} + \frac {3 b^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{2} + 6 b^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} + \frac {3 b^{2} c^{2} x \cos ^{4}{\left (d + e x \right )}}{4} + 6 b^{2} c^{2} x \cos ^{2}{\left (d + e x \right )} + 2 b^{2} c^{2} x + \frac {3 b^{2} c^{2} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{4 e} - \frac {3 b^{2} c^{2} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{4 e} - \frac {4 b^{2} c \sqrt {b^{2} + c^{2}} \cos ^{3}{\left (d + e x \right )}}{e} - \frac {4 b^{2} c \sqrt {b^{2} + c^{2}} \cos {\left (d + e x \right )}}{e} + \frac {b c^{3} \sin ^{4}{\left (d + e x \right )}}{e} - \frac {6 b c^{3} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {4 b c^{2} \sqrt {b^{2} + c^{2}} \sin ^{3}{\left (d + e x \right )}}{e} + \frac {4 b c^{2} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )}}{e} + \frac {3 c^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 c^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + 3 c^{4} x \sin ^{2}{\left (d + e x \right )} + \frac {3 c^{4} x \cos ^{4}{\left (d + e x \right )}}{8} + 3 c^{4} x \cos ^{2}{\left (d + e x \right )} + c^{4} x - \frac {5 c^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} - \frac {3 c^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} - \frac {3 c^{4} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {4 c^{3} \sqrt {b^{2} + c^{2}} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {8 c^{3} \sqrt {b^{2} + c^{2}} \cos ^{3}{\left (d + e x \right )}}{3 e} - \frac {4 c^{3} \sqrt {b^{2} + c^{2}} \cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (b \cos {\relax (d )} + c \sin {\relax (d )} + \sqrt {b^{2} + c^{2}}\right )^{4} & \text {otherwise} \end {cases} \]

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 - b**3*c*cos(d + e*x)**4/e - 6
*b**3*c*cos(d + e*x)**2/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 + b*c**3*sin(d + e*x)**4/e - 6*b*c**3*cos(d + e*x)**2/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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