\(\int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) [489]

Optimal result

Integrand size = 23, antiderivative size = 329 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {32 a \left (a^4-6 a^2 b^2-27 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1155 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1155 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d} \]

[Out]

-2/11*b*cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d+2/231*cos(d*x+c)^3*(a^2+3*b^2+28*a*b*sin(d*x+c))*(a+b*sin(d*x+c)
)^(1/2)/b/d-4/1155*cos(d*x+c)*(4*a^4-21*a^2*b^2-15*b^4-3*a*b*(a^2+31*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b
^3/d+32/1155*a*(a^4-6*a^2*b^2-27*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^4/d/((a+b*sin(d*x+c))/(a+b))^(1/2)
-8/1155*(4*a^6-25*a^4*b^2+6*a^2*b^4+15*b^6)*(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)*Elli
pticF(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^4/d/(a+b*sin(d*x+c))
^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2771, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{231 b d}-\frac {32 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1155 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{1155 b^3 d}+\frac {8 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{1155 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d} \]

[In]

Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(-2*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) - (32*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(c - Pi/2 +
d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^6
- 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a +
b)])/(1155*b^4*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(a^2 + 3*b^2 + 28*a*b
*Sin[c + d*x]))/(231*b*d) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^4 - 21*a^2*b^2 - 15*b^4 - 3*a*b*(a^2
+ 31*b^2)*Sin[c + d*x]))/(1155*b^3*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 2771

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

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*} \text {integral}& = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2}{11} \int \frac {\cos ^4(c+d x) \left (\frac {11 a^2}{2}+\frac {b^2}{2}+6 a b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}+\frac {8 \int \frac {\cos ^2(c+d x) \left (\frac {3}{4} b^2 \left (29 a^2+3 b^2\right )+\frac {3}{4} a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{231 b^2} \\ & = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d}+\frac {32 \int \frac {-\frac {3}{8} b^2 \left (a^4-114 a^2 b^2-15 b^4\right )-\frac {3}{2} a b \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^4} \\ & = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d}-\frac {\left (16 a \left (a^4-6 a^2 b^2-27 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{1155 b^4}+\frac {\left (4 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{1155 b^4} \\ & = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d}-\frac {\left (16 a \left (a^4-6 a^2 b^2-27 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}{1155 b^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\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}{1155 b^4 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {32 a \left (a^4-6 a^2 b^2-27 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)}}{1155 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\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}}}{1155 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.84 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {64 \left (b^2 \left (a^4-114 a^2 b^2-15 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+4 \left (a^5-6 a^3 b^2-27 a b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b (a+b \sin (c+d x)) \left (2 \left (64 a^4-366 a^2 b^2+195 b^4\right ) \cos (c+d x)+5 b^2 \left (-4 a^2+93 b^2\right ) \cos (3 (c+d x))+105 b^4 \cos (5 (c+d x))-16 a b \left (3 a^2+128 b^2\right ) \sin (2 (c+d x))-280 a b^3 \sin (4 (c+d x))\right )}{9240 b^4 d \sqrt {a+b \sin (c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(64*(b^2*(a^4 - 114*a^2*b^2 - 15*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + 4*(a^5 - 6*a^3*b^2 - 2
7*a*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(
a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*(a + b*Sin[c + d*x])*(2*(64*a^4 - 366*a^2*b^2 + 195*b^4)*Cos[
c + d*x] + 5*b^2*(-4*a^2 + 93*b^2)*Cos[3*(c + d*x)] + 105*b^4*Cos[5*(c + d*x)] - 16*a*b*(3*a^2 + 128*b^2)*Sin[
2*(c + d*x)] - 280*a*b^3*Sin[4*(c + d*x)]))/(9240*b^4*d*Sqrt[a + b*Sin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1354\) vs. \(2(371)=742\).

Time = 3.72 (sec) , antiderivative size = 1355, normalized size of antiderivative = 4.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1355\)

[In]

int(cos(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/1155*(60*b^7*sin(d*x+c)+60*((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))*b^7-16*((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^7+16*((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^6*b-12*((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^2-100*((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^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)*EllipticF(((a+b*sin(d*x+c))/(a-b)
)^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4+24*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+si
n(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^2*b^5+372*((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^6+112*((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^5*b^2+336*((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^3*b^4-432*((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*b^6-245
*a*b^6*sin(d*x+c)^6-145*a^2*b^5*sin(d*x+c)^5+a^3*b^4*sin(d*x+c)^4+766*a*b^6*sin(d*x+c)^4-2*a^4*b^3*sin(d*x+c)^
3+518*a^2*b^5*sin(d*x+c)^3-8*a^5*b^2*sin(d*x+c)^2+46*a^3*b^4*sin(d*x+c)^2-581*a*b^6*sin(d*x+c)^2-105*b^7*sin(d
*x+c)^7+300*b^7*sin(d*x+c)^5-255*b^7*sin(d*x+c)^3+8*b^2*a^5-47*b^4*a^3+60*b^6*a+2*a^4*b^3*sin(d*x+c)-373*a^2*b
^5*sin(d*x+c))/b^5/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.77 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (8 \, a^{6} - 51 \, a^{4} b^{2} + 126 \, a^{2} b^{4} + 45 \, b^{6}\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 ) + 2 \, \sqrt {2} {\left (8 \, a^{6} - 51 \, a^{4} b^{2} + 126 \, a^{2} b^{4} + 45 \, b^{6}\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 ) - 24 \, \sqrt {2} {\left (-i \, a^{5} b + 6 i \, a^{3} b^{3} + 27 i \, a b^{5}\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 ) - 24 \, \sqrt {2} {\left (i \, a^{5} b - 6 i \, a^{3} b^{3} - 27 i \, a b^{5}\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 (105 \, b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (4 \, a^{4} b^{2} - 21 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right ) - 2 \, {\left (70 \, a b^{5} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b^{3} + 31 \, 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 )}}{3465 \, b^{5} d} \]

[In]

integrate(cos(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

2/3465*(2*sqrt(2)*(8*a^6 - 51*a^4*b^2 + 126*a^2*b^4 + 45*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) + 2*sqrt(2)*(8
*a^6 - 51*a^4*b^2 + 126*a^2*b^4 + 45*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) - 24*sqrt(2)*(-I*a^5*b + 6*I*a^3
*b^3 + 27*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, weiers
trassPInverse(-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)) - 24*sqrt(2)*(I*a^5*b - 6*I*a^3*b^3 - 27*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*(105*b^6*cos(d*x + c)^5 - 5*(a^2*b
^4 + 3*b^6)*cos(d*x + c)^3 + 2*(4*a^4*b^2 - 21*a^2*b^4 - 15*b^6)*cos(d*x + c) - 2*(70*a*b^5*cos(d*x + c)^3 + 3
*(a^3*b^3 + 31*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^5*d)

Sympy [F]

\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \cos ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(cos(d*x+c)**4*(a+b*sin(d*x+c))**(3/2),x)

[Out]

Integral((a + b*sin(c + d*x))**(3/2)*cos(c + d*x)**4, x)

Maxima [F]

\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cos(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4, x)

Giac [F]

\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cos(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

int(cos(c + d*x)^4*(a + b*sin(c + d*x))^(3/2),x)

[Out]

int(cos(c + d*x)^4*(a + b*sin(c + d*x))^(3/2), x)