3.604 \(\int \frac{\cos (c+d x) (1-\cos ^2(c+d x))}{(a+b \cos (c+d x))^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}+\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d} \]

[Out]

(2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b
]*d) - Sin[c + d*x]/(b^2*d) - (a*Sin[c + d*x])/(b^2*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 0.258268, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3032, 3023, 2735, 2659, 205} \[ -\frac{2 \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}+\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b
]*d) - Sin[c + d*x]/(b^2*d) - (a*Sin[c + d*x])/(b^2*d*(a + b*Cos[c + d*x]))

Rule 3032

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e
_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1
))/(b^2*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b
*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d)) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f
*x] + b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}-\frac{\int \frac{-b \left (a^2-b^2\right )-a \left (a^2-b^2\right ) \cos (c+d x)+b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=-\frac{\sin (c+d x)}{b^2 d}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}-\frac{\int \frac{-b^2 \left (a^2-b^2\right )-2 a b \left (a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}-\frac{\left (2 a^2-b^2\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^3}\\ &=\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}-\frac{\left (2 \left (2 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^3 d}\\ &=\frac{2 a x}{b^3}-\frac{2 \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^3 \sqrt{a+b} d}-\frac{\sin (c+d x)}{b^2 d}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.999303, size = 132, normalized size = 1.18 \[ \frac{\frac{4 a^2 c+4 a^2 d x-4 a b \sin (c+d x)+4 a b (c+d x) \cos (c+d x)-b^2 \sin (2 (c+d x))}{a+b \cos (c+d x)}+\frac{4 \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}}{2 b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(2*a^2 - b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (4*a^2*c + 4*a^2*d*
x + 4*a*b*(c + d*x)*Cos[c + d*x] - 4*a*b*Sin[c + d*x] - b^2*Sin[2*(c + d*x)])/(a + b*Cos[c + d*x]))/(2*b^3*d)

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Maple [A]  time = 0.032, size = 198, normalized size = 1.8 \begin{align*} -2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{b}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}+4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) a}{d{b}^{3}}}-2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) a}{d{b}^{2} \left ( a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }}-4\,{\frac{{a}^{2}}{d{b}^{3}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{1}{db\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/b^2*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)+4/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a-2/d/b^2*tan(1/2*d*x+
1/2*c)*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)-4/d*a^2/b^3/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(
1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+2/d/b/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1
/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60559, size = 1002, normalized size = 8.95 \begin{align*} \left [\frac{4 \,{\left (a^{3} b - a b^{3}\right )} d x \cos \left (d x + c\right ) + 4 \,{\left (a^{4} - a^{2} b^{2}\right )} d x +{\left (2 \, a^{3} - a b^{2} +{\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \,{\left (2 \, a^{3} b - 2 \, a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left ({\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right ) +{\left (a^{3} b^{3} - a b^{5}\right )} d\right )}}, \frac{2 \,{\left (a^{3} b - a b^{3}\right )} d x \cos \left (d x + c\right ) + 2 \,{\left (a^{4} - a^{2} b^{2}\right )} d x -{\left (2 \, a^{3} - a b^{2} +{\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) -{\left (2 \, a^{3} b - 2 \, a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right ) +{\left (a^{3} b^{3} - a b^{5}\right )} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(4*(a^3*b - a*b^3)*d*x*cos(d*x + c) + 4*(a^4 - a^2*b^2)*d*x + (2*a^3 - a*b^2 + (2*a^2*b - b^3)*cos(d*x +
c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(d*x +
c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - 2*(2*a^3*b - 2*a*b^3 +
(a^2*b^2 - b^4)*cos(d*x + c))*sin(d*x + c))/((a^2*b^4 - b^6)*d*cos(d*x + c) + (a^3*b^3 - a*b^5)*d), (2*(a^3*b
- a*b^3)*d*x*cos(d*x + c) + 2*(a^4 - a^2*b^2)*d*x - (2*a^3 - a*b^2 + (2*a^2*b - b^3)*cos(d*x + c))*sqrt(a^2 -
b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (2*a^3*b - 2*a*b^3 + (a^2*b^2 - b^4)*cos(d
*x + c))*sin(d*x + c))/((a^2*b^4 - b^6)*d*cos(d*x + c) + (a^3*b^3 - a*b^5)*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(cos(d*x+c)*(1-cos(d*x+c)**2)/(a+b*cos(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.54517, size = 279, normalized size = 2.49 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )} a}{b^{3}} + \frac{{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}{\left (2 \, a^{2} - b^{2}\right )}}{\sqrt{a^{2} - b^{2}} b^{3}} - \frac{2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b\right )} b^{2}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((d*x + c)*a/b^3 + (pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*t
an(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))*(2*a^2 - b^2)/(sqrt(a^2 - b^2)*b^3) - (2*a*tan(1/2*d*x + 1/2*c)^3 - b*t
an(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c) + b*tan(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^4 - b*tan(
1/2*d*x + 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)*b^2))/d