3.6.34 \(\int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [534]

Optimal. Leaf size=293 \[ -\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}+\frac {16 a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 \left (32 a^4-37 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d} \]

[Out]

-2/3*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))^(3/2)-20/21*cos(d*x+c)^3*(8*a+b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^(1/2
)+8/21*cos(d*x+c)*(32*a^2-5*b^2-24*a*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^5/d-16/21*a*(32*a^2-29*b^2)*(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)+16/21*(32*a^4-37*a^2*b^2+5*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 = 0.34, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2772, 2942, 2944, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {16 a \left (32 a^2-29 b^2\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 )}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{21 b^5 d}-\frac {16 \left (32 a^4-37 a^2 b^2+5 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 )}{21 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*Cos[c + d*x]^5)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) + (16*a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2,
(2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(21*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*(32*a^4 - 37*a^2
*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(21*b^6*d*Sqrt[
a + b*Sin[c + d*x]]) - (20*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(21*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (8*Cos
[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d*x]))/(21*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 2942

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 + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + 1)*(m + p + 1
))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

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))^{5/2}} \, dx &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {10 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx}{3 b}\\ &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \int \frac {\cos ^2(c+d x) \left (-\frac {b}{2}-4 a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{7 b^3}\\ &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d}+\frac {32 \int \frac {\frac {1}{4} b \left (8 a^2-5 b^2\right )+\frac {1}{4} a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^5}\\ &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d}+\frac {\left (8 a \left (32 a^2-29 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{21 b^6}-\frac {\left (8 \left (32 a^4-37 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^6}\\ &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d}+\frac {\left (8 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{21 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 \left (32 a^4-37 a^2 b^2+5 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}{21 b^6 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}+\frac {16 a \left (32 a^2-29 b^2\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)}}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 \left (32 a^4-37 a^2 b^2+5 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}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d}\\ \end {align*}

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Mathematica [A]
time = 1.22, size = 244, normalized size = 0.83 \begin {gather*} \frac {-32 (a+b) \left (a \left (32 a^3+32 a^2 b-29 a b^2-29 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+\left (-32 a^4+37 a^2 b^2-5 b^4\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}+\frac {1}{2} b \cos (c+d x) \left (1024 a^4-736 a^2 b^2-111 b^4+\left (-64 a^2 b^2+52 b^4\right ) \cos (2 (c+d x))+3 b^4 \cos (4 (c+d x))+1280 a^3 b \sin (c+d x)-1076 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))\right )}{42 b^6 d (a+b \sin (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-32*(a + b)*(a*(32*a^3 + 32*a^2*b - 29*a*b^2 - 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (-32
*a^4 + 37*a^2*b^2 - 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/
2) + (b*Cos[c + d*x]*(1024*a^4 - 736*a^2*b^2 - 111*b^4 + (-64*a^2*b^2 + 52*b^4)*Cos[2*(c + d*x)] + 3*b^4*Cos[4
*(c + d*x)] + 1280*a^3*b*Sin[c + d*x] - 1076*a*b^3*Sin[c + d*x] + 12*a*b^3*Sin[3*(c + d*x)]))/2)/(42*b^6*d*(a
+ b*Sin[c + d*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1641\) vs. \(2(335)=670\).
time = 2.23, size = 1642, normalized size = 5.60

method	result	size
default	\(\text {Expression too large to display}\)	\(1642\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/21*(6*a*b^5*sin(d*x+c)*cos(d*x+c)^4+(160*a^3*b^3-136*a*b^5)*cos(d*x+c)^2*sin(d*x+c)+8*(-b/(a-b)*sin(d*x+c)-b
/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*b*(32*EllipticF((b/(a-b
)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^4*b-24*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a
-b)/(a+b))^(1/2))*a^3*b^2-37*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^3+24*El
lipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a*b^4+5*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b
)*a)^(1/2),((a-b)/(a+b))^(1/2))*b^5-32*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^5
+61*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2-29*EllipticE((b/(a-b)*sin(d*x+
c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a*b^4)*sin(d*x+c)+3*b^6*cos(d*x+c)^6+(-16*a^2*b^4+10*b^6)*cos(d*x+c)^
4+(128*a^4*b^2-84*a^2*b^4-20*b^6)*cos(d*x+c)^2+256*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/
(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+1/(a-b)*a
)^(1/2)*a^5*b-192*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*s
in(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*a^4*b^2-296*(-b/(a+b)*sin
(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-
b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*a^3*b^3+192*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b
)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d
*x+c)+1/(a-b)*a)^(1/2)*a^2*b^4+40*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*Elli
pticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*a*b^5-256
*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(
a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*a^6+488*EllipticE((b/(a-b)*sin(d*x+c)+1/(a
-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(
a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*a^4*b^2-232*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2
))*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2
)*a^2*b^4)/(a+b*sin(d*x+c))^(3/2)/b^7/cos(d*x+c)/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))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^6/(b*sin(d*x + c) + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 874, normalized size = 2.98 \begin {gather*} -\frac {2 \, {\left (4 \, {\left (\sqrt {2} {\left (64 \, a^{4} b^{2} - 82 \, a^{2} b^{4} + 15 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (64 \, a^{5} b - 82 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (64 \, a^{6} - 18 \, a^{4} b^{2} - 67 \, a^{2} b^{4} + 15 \, b^{6}\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 ) + 4 \, {\left (\sqrt {2} {\left (64 \, a^{4} b^{2} - 82 \, a^{2} b^{4} + 15 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (64 \, a^{5} b - 82 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (64 \, a^{6} - 18 \, a^{4} b^{2} - 67 \, a^{2} b^{4} + 15 \, b^{6}\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^{3} b^{3} + 29 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (32 i \, a^{4} b^{2} - 29 i \, a^{2} b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{5} b + 3 i \, a^{3} b^{3} - 29 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^{3} b^{3} - 29 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-32 i \, a^{4} b^{2} + 29 i \, a^{2} b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{5} b - 3 i \, a^{3} b^{3} + 29 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 (3 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (8 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (32 \, a^{4} b^{2} - 21 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (3 \, a b^{5} \cos \left (d x + c\right )^{3} + 4 \, {\left (20 \, a^{3} b^{3} - 17 \, 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 )}}{63 \, {\left (b^{9} d \cos \left (d x + c\right )^{2} - 2 \, a b^{8} d \sin \left (d x + c\right ) - {\left (a^{2} b^{7} + b^{9}\right )} 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))^(5/2),x, algorithm="fricas")

[Out]

-2/63*(4*(sqrt(2)*(64*a^4*b^2 - 82*a^2*b^4 + 15*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(64*a^5*b - 82*a^3*b^3 + 15*a*
b^5)*sin(d*x + c) - sqrt(2)*(64*a^6 - 18*a^4*b^2 - 67*a^2*b^4 + 15*b^6))*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) + 4
*(sqrt(2)*(64*a^4*b^2 - 82*a^2*b^4 + 15*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(64*a^5*b - 82*a^3*b^3 + 15*a*b^5)*sin
(d*x + c) - sqrt(2)*(64*a^6 - 18*a^4*b^2 - 67*a^2*b^4 + 15*b^6))*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*(sqr
t(2)*(-32*I*a^3*b^3 + 29*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)*(32*I*a^4*b^2 - 29*I*a^2*b^4)*sin(d*x + c) + sqrt
(2)*(32*I*a^5*b + 3*I*a^3*b^3 - 29*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*co
s(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) - 12*(sqrt(2)*(32*I*a^3*b^3 - 29*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt
(2)*(-32*I*a^4*b^2 + 29*I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(-32*I*a^5*b - 3*I*a^3*b^3 + 29*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)) + 3*(3*
b^6*cos(d*x + c)^5 - 2*(8*a^2*b^4 - 5*b^6)*cos(d*x + c)^3 + 4*(32*a^4*b^2 - 21*a^2*b^4 - 5*b^6)*cos(d*x + c) +
2*(3*a*b^5*cos(d*x + c)^3 + 4*(20*a^3*b^3 - 17*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(
b^9*d*cos(d*x + c)^2 - 2*a*b^8*d*sin(d*x + c) - (a^2*b^7 + b^9)*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))**(5/2),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))^(5/2),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 )}^{5/2}} \,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))^(5/2),x)

[Out]

int(cos(c + d*x)^6/(a + b*sin(c + d*x))^(5/2), x)

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