3.421 \(\int (a+b \cos (c+d x))^2 \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d} \]

[Out]

((2*a^2 + b^2)*x)/2 + (2*a*b*Sin[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.0149788, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2644} \[ \frac{1}{2} x \left (2 a^2+b^2\right )+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^2,x]

[Out]

((2*a^2 + b^2)*x)/2 + (2*a*b*Sin[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)

Rule 2644

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[((2*a^2 + b^2)*x)/2, x] + (-Simp[(2*a*b*Cos[c
+ d*x])/d, x] - Simp[(b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), x]) /; FreeQ[{a, b, c, d}, x]

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^2 \, dx &=\frac{1}{2} \left (2 a^2+b^2\right ) x+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0716666, size = 46, normalized size = 0.92 \[ \frac{2 \left (2 a^2+b^2\right ) (c+d x)+8 a b \sin (c+d x)+b^2 \sin (2 (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^2,x]

[Out]

(2*(2*a^2 + b^2)*(c + d*x) + 8*a*b*Sin[c + d*x] + b^2*Sin[2*(c + d*x)])/(4*d)

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Maple [A]  time = 0.031, size = 51, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,ab\sin \left ( dx+c \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^2,x)

[Out]

1/d*(b^2*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+2*a*b*sin(d*x+c)+a^2*(d*x+c))

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Maxima [A]  time = 0.954685, size = 59, normalized size = 1.18 \begin{align*} a^{2} x + \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2}}{4 \, d} + \frac{2 \, a b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

a^2*x + 1/4*(2*d*x + 2*c + sin(2*d*x + 2*c))*b^2/d + 2*a*b*sin(d*x + c)/d

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Fricas [A]  time = 1.80501, size = 93, normalized size = 1.86 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d x +{\left (b^{2} \cos \left (d x + c\right ) + 4 \, a b\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

1/2*((2*a^2 + b^2)*d*x + (b^2*cos(d*x + c) + 4*a*b)*sin(d*x + c))/d

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Sympy [A]  time = 0.281447, size = 78, normalized size = 1.56 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b \sin{\left (c + d x \right )}}{d} + \frac{b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**2,x)

[Out]

Piecewise((a**2*x + 2*a*b*sin(c + d*x)/d + b**2*x*sin(c + d*x)**2/2 + b**2*x*cos(c + d*x)**2/2 + b**2*sin(c +
d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(a + b*cos(c))**2, True))

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Giac [A]  time = 1.52084, size = 58, normalized size = 1.16 \begin{align*} \frac{1}{2} \,{\left (2 \, a^{2} + b^{2}\right )} x + \frac{b^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{2 \, a b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(2*a^2 + b^2)*x + 1/4*b^2*sin(2*d*x + 2*c)/d + 2*a*b*sin(d*x + c)/d