3.438 \(\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=120 \[ \frac{\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac{\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}-\frac{4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d} \]

[Out]

(-4*a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) - (a*Sin[c + d*x]^
2)/(b^3*d) + Sin[c + d*x]^3/(3*b^2*d) - (a^2 - b^2)^2/(b^5*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 0.101275, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2668, 697} \[ \frac{\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac{\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}-\frac{4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]

[Out]

(-4*a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) - (a*Sin[c + d*x]^
2)/(b^3*d) + Sin[c + d*x]^3/(3*b^2*d) - (a^2 - b^2)^2/(b^5*d*(a + b*Sin[c + d*x]))

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a^2 \left (1-\frac{2 b^2}{3 a^2}\right )-2 a x+x^2+\frac{\left (a^2-b^2\right )^2}{(a+x)^2}-\frac{4 \left (a^3-a b^2\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac{\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d}-\frac{\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.634727, size = 127, normalized size = 1.06 \[ \frac{\left (8 a^2 b-4 b^3\right ) \sin (c+d x)+\frac{b^4 \cos ^4(c+d x)-4 \left (a^2-b^2\right ) \left (3 a^2 \log (a+b \sin (c+d x))+a^2+3 a b \sin (c+d x) \log (a+b \sin (c+d x))-b^2\right )}{a+b \sin (c+d x)}-2 a b^2 \sin ^2(c+d x)}{3 b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]

[Out]

((8*a^2*b - 4*b^3)*Sin[c + d*x] - 2*a*b^2*Sin[c + d*x]^2 + (b^4*Cos[c + d*x]^4 - 4*(a^2 - b^2)*(a^2 - b^2 + 3*
a^2*Log[a + b*Sin[c + d*x]] + 3*a*b*Log[a + b*Sin[c + d*x]]*Sin[c + d*x]))/(a + b*Sin[c + d*x]))/(3*b^5*d)

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Maple [A]  time = 0.079, size = 174, normalized size = 1.5 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{{b}^{3}d}}+3\,{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d{b}^{4}}}-2\,{\frac{\sin \left ( dx+c \right ) }{{b}^{2}d}}-4\,{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{5}}}+4\,{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{3}d}}-{\frac{{a}^{4}}{d{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+2\,{\frac{{a}^{2}}{{b}^{3}d \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{1}{bd \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*sin(d*x+c))^2,x)

[Out]

1/3*sin(d*x+c)^3/b^2/d-a*sin(d*x+c)^2/b^3/d+3/d/b^4*a^2*sin(d*x+c)-2*sin(d*x+c)/b^2/d-4/d*a^3/b^5*ln(a+b*sin(d
*x+c))+4*a*ln(a+b*sin(d*x+c))/b^3/d-1/d/b^5/(a+b*sin(d*x+c))*a^4+2/d/b^3/(a+b*sin(d*x+c))*a^2-1/b/d/(a+b*sin(d
*x+c))

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Maxima [A]  time = 0.953096, size = 157, normalized size = 1.31 \begin{align*} -\frac{\frac{3 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}{b^{6} \sin \left (d x + c\right ) + a b^{5}} - \frac{b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 3 \,{\left (3 \, a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(3*(a^4 - 2*a^2*b^2 + b^4)/(b^6*sin(d*x + c) + a*b^5) - (b^2*sin(d*x + c)^3 - 3*a*b*sin(d*x + c)^2 + 3*(3
*a^2 - 2*b^2)*sin(d*x + c))/b^4 + 12*(a^3 - a*b^2)*log(b*sin(d*x + c) + a)/b^5)/d

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Fricas [A]  time = 2.81643, size = 359, normalized size = 2.99 \begin{align*} \frac{2 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \, a^{4} + 27 \, a^{2} b^{2} - 16 \, b^{4} - 4 \,{\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 24 \,{\left (a^{4} - a^{2} b^{2} +{\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) +{\left (4 \, a b^{3} \cos \left (d x + c\right )^{2} + 18 \, a^{3} b - 13 \, a b^{3}\right )} \sin \left (d x + c\right )}{6 \,{\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/6*(2*b^4*cos(d*x + c)^4 - 6*a^4 + 27*a^2*b^2 - 16*b^4 - 4*(3*a^2*b^2 - 2*b^4)*cos(d*x + c)^2 - 24*(a^4 - a^2
*b^2 + (a^3*b - a*b^3)*sin(d*x + c))*log(b*sin(d*x + c) + a) + (4*a*b^3*cos(d*x + c)^2 + 18*a^3*b - 13*a*b^3)*
sin(d*x + c))/(b^6*d*sin(d*x + c) + 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)**5/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.12152, size = 203, normalized size = 1.69 \begin{align*} -\frac{\frac{12 \,{\left (a^{3} - a b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}} - \frac{b^{4} \sin \left (d x + c\right )^{3} - 3 \, a b^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} \sin \left (d x + c\right ) - 6 \, b^{4} \sin \left (d x + c\right )}{b^{6}} - \frac{3 \,{\left (4 \, a^{3} b \sin \left (d x + c\right ) - 4 \, a b^{3} \sin \left (d x + c\right ) + 3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{5}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*(12*(a^3 - a*b^2)*log(abs(b*sin(d*x + c) + a))/b^5 - (b^4*sin(d*x + c)^3 - 3*a*b^3*sin(d*x + c)^2 + 9*a^2
*b^2*sin(d*x + c) - 6*b^4*sin(d*x + c))/b^6 - 3*(4*a^3*b*sin(d*x + c) - 4*a*b^3*sin(d*x + c) + 3*a^4 - 2*a^2*b
^2 - b^4)/((b*sin(d*x + c) + a)*b^5))/d