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

Optimal. Leaf size=118 \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d} \]

[Out]

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

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Rubi [A]  time = 0.105313, antiderivative size = 118, 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} \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.195237, size = 103, normalized size = 0.87 \[ \frac{6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)-12 a b \left (a^2-2 b^2\right ) \sin (c+d x)+12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-4 a b^3 \sin ^3(c+d x)+3 b^4 \cos ^4(c+d x)}{12 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^4*Cos[c + d*x]^4 + 12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] - 12*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 6*b^2*(
a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^3*Sin[c + d*x]^3)/(12*b^5*d)

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Maple [A]  time = 0.036, size = 163, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,bd}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d{b}^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{bd}}-{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d{b}^{4}}}+2\,{\frac{a\sin \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{4}}{d{b}^{5}}}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}} \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)),x)

[Out]

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

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Maxima [A]  time = 0.963665, size = 146, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \,{\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \,{\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{12 \, 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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.71323, size = 250, normalized size = 2.12 \begin{align*} \frac{3 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \,{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} 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)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.125, size = 162, normalized size = 1.37 \begin{align*} \frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b \sin \left (d x + c\right )^{2} - 12 \, b^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) + 24 \, a b^{2} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, 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)),x, algorithm="giac")

[Out]

1/12*((3*b^3*sin(d*x + c)^4 - 4*a*b^2*sin(d*x + c)^3 + 6*a^2*b*sin(d*x + c)^2 - 12*b^3*sin(d*x + c)^2 - 12*a^3
*sin(d*x + c) + 24*a*b^2*sin(d*x + c))/b^4 + 12*(a^4 - 2*a^2*b^2 + b^4)*log(abs(b*sin(d*x + c) + a))/b^5)/d