3.6.22 \(\int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [522]

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

[Out]

-2*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))^(1/2)+20/63*cos(d*x+c)^3*(8*a-7*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^3/
d-8/63*cos(d*x+c)*(a*(32*a^2-33*b^2)-3*b*(8*a^2-7*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^5/d+16/63*(32*a^4-
57*a^2*b^2+21*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/
2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^6/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-16/63*a*(32*a^4-65
*a^2*b^2+33*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*
d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^6/d/(a+b*sin(d*x+c))^(1/2)

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Rubi [A]
time = 0.35, antiderivative size = 313, normalized size of antiderivative = 1.00, 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 = {2772, 2944, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\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 (32 a^4-65 a^2 b^2+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 (32 a^4-57 a^2 b^2+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)}} \end {gather*}

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 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2772

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[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 2831

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 2944

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]

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*}

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Mathematica [A]
time = 1.50, size = 273, normalized size = 0.87 \begin {gather*} \frac {64 \left (32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-1024 a^4+1760 a^2 b^2-595 b^4+\left (-64 a^2 b^2+84 b^4\right ) \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))\right )}{252 b^6 d \sqrt {a+b \sin (c+d x)}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1194\) vs. \(2(357)=714\).
time = 2.26, size = 1195, normalized size = 3.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1195\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.21, size = 723, normalized size = 2.31 \begin {gather*} \frac {2 \, {\left (8 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} b^{4}\right )}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 8 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} b^{4}\right )}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 12 \, {\left (\sqrt {2} {\left (-32 i \, a^{4} b^{2} + 57 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{5} b + 57 i \, a^{3} b^{3} - 21 i \, a b^{5}\right )}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 12 \, {\left (\sqrt {2} {\left (32 i \, a^{4} b^{2} - 57 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{5} b - 57 i \, a^{3} b^{3} + 21 i \, a b^{5}\right )}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (7 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (8 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (32 \, a^{4} b^{2} - 57 \, a^{2} b^{4} + 21 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, a b^{5} \cos \left (d x + c\right )^{3} - 8 \, {\left (2 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{189 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \end {gather*}

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]

2/189*(8*(sqrt(2)*(32*a^5*b - 69*a^3*b^3 + 39*a*b^5)*sin(d*x + c) + sqrt(2)*(32*a^6 - 69*a^4*b^2 + 39*a^2*b^4)
)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x +
c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + 8*(sqrt(2)*(32*a^5*b - 69*a^3*b^3 + 39*a*b^5)*sin(d*x + c) + sqrt(2)*(32
*a^6 - 69*a^4*b^2 + 39*a^2*b^4))*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*
I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) - 12*(sqrt(2)*(-32*I*a^4*b^2 + 57*I*a^2*b
^4 - 21*I*b^6)*sin(d*x + c) + sqrt(2)*(-32*I*a^5*b + 57*I*a^3*b^3 - 21*I*a*b^5))*sqrt(I*b)*weierstrassZeta(-4/
3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8
*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) - 12*(sqrt(2)*(32*I*a^4*b^2 -
57*I*a^2*b^4 + 21*I*b^6)*sin(d*x + c) + sqrt(2)*(32*I*a^5*b - 57*I*a^3*b^3 + 21*I*a*b^5))*sqrt(-I*b)*weierstr
assZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b
^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) + 3*(7*b^6*cos(d
*x + c)^5 - 2*(8*a^2*b^4 - 7*b^6)*cos(d*x + c)^3 - 4*(32*a^4*b^2 - 57*a^2*b^4 + 21*b^6)*cos(d*x + c) + 2*(5*a*
b^5*cos(d*x + c)^3 - 8*(2*a^3*b^3 - 3*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^8*d*sin(
d*x + c) + a*b^7*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^6}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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