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

Optimal. Leaf size=32 \[ -\frac{A \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0499524, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {3011} \[ -\frac{A \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(1 + 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 (A-\frac{A (2+m) \cos ^2(c+d x)}{1+m}\right ) \, dx &=-\frac{A (b \cos (c+d x))^{1+m} \sin (c+d x)}{b d (1+m)}\\ \end{align*}

Mathematica [C]  time = 0.197603, size = 119, normalized size = 3.72 \[ \frac{A \sin (c+d x) \cos (c+d x) (b \cos (c+d x))^m \left ((m+2) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right )-(m+3) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )\right )}{d (m+1) (m+3) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*Cos[c + d*x]*(b*Cos[c + d*x])^m*(-((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])*Sin[c + d*x])/(d*(1 + m)*
(3 + m)*Sqrt[Sin[c + d*x]^2])

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Maple [F]  time = 1.523, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{m} \left ( A-{\frac{A \left ( 2+m \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{1+m}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.15901, size = 236, normalized size = 7.38 \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} A 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} A 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 + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.32814, size = 80, normalized size = 2.5 \begin{align*} -\frac{\left (b \cos \left (d x + c\right )\right )^{m} A \cos \left (d x + c\right ) \sin \left (d x + c\right )}{d m + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError