3.35 \(\int (b \cos (c+d x))^m (-\frac{C (1+m)}{2+m}+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=31 \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)} \]

[Out]

(C*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(2 + m))

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Rubi [A]  time = 0.0416858, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3011} \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x])^m*(-((C*(1 + m))/(2 + m)) + C*Cos[c + d*x]^2),x]

[Out]

(C*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(2 + m))

Rule 3011

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

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

Mathematica [C]  time = 0.193215, size = 113, normalized size = 3.65 \[ \frac{C \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^m \left ((m+3) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )-(m+2) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right )\right )}{d (m+2) (m+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x])^m*(-((C*(1 + m))/(2 + m)) + C*Cos[c + d*x]^2),x]

[Out]

(C*(b*Cos[c + d*x])^m*Cot[c + d*x]*((3 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2] - (2
+ m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2
+ m)*(3 + m))

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Maple [F]  time = 1.934, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{m} \left ( -{\frac{C \left ( 1+m \right ) }{2+m}}+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(d*x+c))^m*(-C*(1+m)/(2+m)+C*cos(d*x+c)^2),x)

[Out]

int((b*cos(d*x+c))^m*(-C*(1+m)/(2+m)+C*cos(d*x+c)^2),x)

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Maxima [B]  time = 2.21787, size = 236, normalized size = 7.61 \begin{align*} -\frac{{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac{1}{2} \, m} C b^{m} \sin \left (-{\left (d x + c\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) -{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac{1}{2} \, m} C b^{m} \sin \left (-{\left (d x + c\right )}{\left (m - 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )}{4 \cdot 2^{m} d{\left (m + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^m*(-C*(1+m)/(2+m)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/4*((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/2*m)*C*b^m*sin(-(d*x + c)*(m + 2)
+ m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x
+ 2*c) + 1)^(1/2*m)*C*b^m*sin(-(d*x + c)*(m - 2) + m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(2^m*d*
(m + 2))

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Fricas [A]  time = 1.39454, size = 81, normalized size = 2.61 \begin{align*} \frac{\left (b \cos \left (d x + c\right )\right )^{m} C \cos \left (d x + c\right ) \sin \left (d x + c\right )}{d m + 2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^m*(-C*(1+m)/(2+m)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

(b*cos(d*x + c))^m*C*cos(d*x + c)*sin(d*x + c)/(d*m + 2*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))**m*(-C*(1+m)/(2+m)+C*cos(d*x+c)**2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^m*(-C*(1+m)/(2+m)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError