3.522 \(\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=313 \[ -\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}+\frac{16 a \left (-65 a^2 b^2+32 a^4+33 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 \left (-57 a^2 b^2+32 a^4+21 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}} \]

[Out]

(-2*Cos[c + d*x]^5)/(b*d*Sqrt[a + b*Sin[c + d*x]]) + (20*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Si
n[c + d*x]])/(63*b^3*d) - (16*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt
[a + b*Sin[c + d*x]])/(63*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*Ell
ipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(63*b^6*d*Sqrt[a + b*Sin[c + d*x
]]) - (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*b
^5*d)

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Rubi [A]  time = 0.539254, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2693, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}+\frac{16 a \left (-65 a^2 b^2+32 a^4+33 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 \left (-57 a^2 b^2+32 a^4+21 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(-2*Cos[c + d*x]^5)/(b*d*Sqrt[a + b*Sin[c + d*x]]) + (20*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Si
n[c + d*x]])/(63*b^3*d) - (16*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt
[a + b*Sin[c + d*x]])/(63*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*Ell
ipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(63*b^6*d*Sqrt[a + b*Sin[c + d*x
]]) - (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*b
^5*d)

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2865

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x]))/(b^2*f*(m + p)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + p)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}-\frac{10 \int \frac{\cos ^4(c+d x) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{b}\\ &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{40 \int \frac{\cos ^2(c+d x) \left (-\frac{a b}{2}-\frac{1}{2} \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{21 b^3}\\ &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac{32 \int \frac{a b \left (2 a^2-3 b^2\right )+\frac{1}{4} \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{63 b^5}\\ &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac{\left (8 \left (32 a^4-57 a^2 b^2+21 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{63 b^6}+\frac{\left (8 a \left (32 a^4-65 a^2 b^2+33 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{63 b^6}\\ &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac{\left (8 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{63 b^6 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (8 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{63 b^6 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{16 \left (32 a^4-57 a^2 b^2+21 b^4\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{16 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}\\ \end{align*}

Mathematica [A]  time = 1.4752, size = 273, normalized size = 0.87 \[ \frac{b \cos (c+d x) \left (\left (84 b^4-64 a^2 b^2\right ) \cos (2 (c+d x))+1760 a^2 b^2-256 a^3 b \sin (c+d x)-1024 a^4+404 a b^3 \sin (c+d x)+20 a b^3 \sin (3 (c+d x))+7 b^4 \cos (4 (c+d x))-595 b^4\right )-64 a \left (-65 a^2 b^2+32 a^4+33 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+64 \left (-57 a^3 b^2-57 a^2 b^3+32 a^4 b+32 a^5+21 a b^4+21 b^5\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{252 b^6 d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(64*(32*a^5 + 32*a^4*b - 57*a^3*b^2 - 57*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(
a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(-2*c + Pi - 2*d*x)
/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-1024*a^4 + 1760*a^2*b^2 - 595*b^4 + (
-64*a^2*b^2 + 84*b^4)*Cos[2*(c + d*x)] + 7*b^4*Cos[4*(c + d*x)] - 256*a^3*b*Sin[c + d*x] + 404*a*b^3*Sin[c + d
*x] + 20*a*b^3*Sin[3*(c + d*x)]))/(252*b^6*d*Sqrt[a + b*Sin[c + d*x]])

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Maple [B]  time = 0.601, size = 1195, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^(3/2),x)

[Out]

-2/63*(7*b^6*sin(d*x+c)^6-10*a*b^5*sin(d*x+c)^5+256*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(
1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b-192*(
(a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*
sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2-520*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a
+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b
^3+360*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Elliptic
F(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4+264*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)
-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2
))*a*b^5-168*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*El
lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^6-256*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+
c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1
/2))*a^6+712*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*El
lipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2-624*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(
d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b)
)^(1/2))*a^2*b^4+168*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^
(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^6+16*a^2*b^4*sin(d*x+c)^4-35*b^6*sin(d*x
+c)^4-32*a^3*b^3*sin(d*x+c)^3+68*a*b^5*sin(d*x+c)^3-128*a^4*b^2*sin(d*x+c)^2+196*a^2*b^4*sin(d*x+c)^2-35*b^6*s
in(d*x+c)^2+32*a^3*b^3*sin(d*x+c)-58*a*b^5*sin(d*x+c)+128*a^4*b^2-212*a^2*b^4+63*b^6)/b^7/cos(d*x+c)/(a+b*sin(
d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^6/(b*sin(d*x + c) + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

integral(-sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^6/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2), x)

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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)**6/(a+b*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^6/(b*sin(d*x + c) + a)^(3/2), x)