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

Optimal. Leaf size=152 \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d} \]

[Out]

(2*(a^2 - b^2)^2*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) +
(4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (2*(a +
b*Sin[c + d*x])^(9/2))/(9*b^5*d)

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Rubi [A]  time = 0.114031, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2668, 697} \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a^2 - b^2)^2*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) +
(4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (2*(a +
b*Sin[c + d*x])^(9/2))/(9*b^5*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)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{\sqrt{a+x}} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2-b^2\right )^2}{\sqrt{a+x}}-4 \left (a^3-a b^2\right ) \sqrt{a+x}+2 \left (3 a^2-b^2\right ) (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{2 \left (a^2-b^2\right )^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}\\ \end{align*}

Mathematica [A]  time = 0.288291, size = 118, normalized size = 0.78 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (-4 \left (48 a^2 b^2-91 b^4\right ) \cos (2 (c+d x))-2496 a^2 b^2-512 a^3 b \sin (c+d x)+1024 a^4+1104 a b^3 \sin (c+d x)+80 a b^3 \sin (3 (c+d x))+35 b^4 \cos (4 (c+d x))+2121 b^4\right )}{1260 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*Sin[c + d*x]]*(1024*a^4 - 2496*a^2*b^2 + 2121*b^4 - 4*(48*a^2*b^2 - 91*b^4)*Cos[2*(c + d*x)] + 35*
b^4*Cos[4*(c + d*x)] - 512*a^3*b*Sin[c + d*x] + 1104*a*b^3*Sin[c + d*x] + 80*a*b^3*Sin[3*(c + d*x)]))/(1260*b^
5*d)

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Maple [A]  time = 0.241, size = 126, normalized size = 0.8 \begin{align*}{\frac{70\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+80\,a{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -96\,{a}^{2}{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+112\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-128\,{a}^{3}b\sin \left ( dx+c \right ) +256\,a{b}^{3}\sin \left ( dx+c \right ) +256\,{a}^{4}-576\,{a}^{2}{b}^{2}+448\,{b}^{4}}{315\,{b}^{5}d}\sqrt{a+b\sin \left ( dx+c \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))^(1/2),x)

[Out]

2/315/b^5*(a+b*sin(d*x+c))^(1/2)*(35*b^4*cos(d*x+c)^4+40*a*b^3*cos(d*x+c)^2*sin(d*x+c)-48*a^2*b^2*cos(d*x+c)^2
+56*b^4*cos(d*x+c)^2-64*a^3*b*sin(d*x+c)+128*a*b^3*sin(d*x+c)+128*a^4-288*a^2*b^2+224*b^4)/d

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Maxima [A]  time = 0.981467, size = 216, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{b \sin \left (d x + c\right ) + a} - \frac{42 \,{\left (3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2}\right )}}{b^{2}} + \frac{35 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{4}}{b^{4}}\right )}}{315 \, b 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))^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(b*sin(d*x + c) + a) - 42*(3*(b*sin(d*x + c) + a)^(5/2) - 10*(b*sin(d*x + c) + a)^(3/2)*a + 15*
sqrt(b*sin(d*x + c) + a)*a^2)/b^2 + (35*(b*sin(d*x + c) + a)^(9/2) - 180*(b*sin(d*x + c) + a)^(7/2)*a + 378*(b
*sin(d*x + c) + a)^(5/2)*a^2 - 420*(b*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(b*sin(d*x + c) + a)*a^4)/b^4)/(b*
d)

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Fricas [A]  time = 2.13668, size = 270, normalized size = 1.78 \begin{align*} \frac{2 \,{\left (35 \, b^{4} \cos \left (d x + c\right )^{4} + 128 \, a^{4} - 288 \, a^{2} b^{2} + 224 \, b^{4} - 8 \,{\left (6 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (5 \, a b^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{315 \, 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))^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*cos(d*x + c)^4 + 128*a^4 - 288*a^2*b^2 + 224*b^4 - 8*(6*a^2*b^2 - 7*b^4)*cos(d*x + c)^2 + 8*(5*a
*b^3*cos(d*x + c)^2 - 8*a^3*b + 16*a*b^3)*sin(d*x + c))*sqrt(b*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))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.2114, size = 217, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (35 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{4} - 126 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} b^{2} + 420 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a b^{2} - 630 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2} b^{2} + 315 \, \sqrt{b \sin \left (d x + c\right ) + a} b^{4}\right )}}{315 \, 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))^(1/2),x, algorithm="giac")

[Out]

2/315*(35*(b*sin(d*x + c) + a)^(9/2) - 180*(b*sin(d*x + c) + a)^(7/2)*a + 378*(b*sin(d*x + c) + a)^(5/2)*a^2 -
420*(b*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(b*sin(d*x + c) + a)*a^4 - 126*(b*sin(d*x + c) + a)^(5/2)*b^2 +
420*(b*sin(d*x + c) + a)^(3/2)*a*b^2 - 630*sqrt(b*sin(d*x + c) + a)*a^2*b^2 + 315*sqrt(b*sin(d*x + c) + a)*b^4
)/(b^5*d)