\(\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) [1160]

Optimal result

Integrand size = 29, antiderivative size = 451 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {8 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{45045 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d} \]

[Out]

-2/39*a*cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2)/d-2/15*cos(d*x+c)^5*(a+b*sin(d*x+c))^(5/2)/d-2/429*(3*a^2+13*b^2)*
cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d-2/9009*cos(d*x+c)^3*(8*a^4-33*a^2*b^2-39*b^4-7*a*b*(a^2+63*b^2)*sin(d*x+
c))*(a+b*sin(d*x+c))^(1/2)/b^2/d+4/45045*cos(d*x+c)*(32*a^6-165*a^4*b^2+450*a^2*b^4+195*b^6-24*a*b*(a^4-5*a^2*
b^2-60*b^4)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^4/d-8/45045*a*(32*a^6-189*a^4*b^2+570*a^2*b^4+1635*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)*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^5/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+8/45045*(32*a^8-197*a^6*b^2+615*a^4*b^4-25
5*a^2*b^6-195*b^8)*(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^5/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2941, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{45045 b^4 d}+\frac {8 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{45045 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d} \]

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(3*a^2 + 13*b^2)*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(429*d) - (2*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x
])^(3/2))/(39*d) - (2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(15*d) + (8*a*(32*a^6 - 189*a^4*b^2 + 570*a^2
*b^4 + 1635*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d*Sqrt[(a +
b*Sin[c + d*x])/(a + b)]) - (8*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(c - Pi
/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Co
s[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^4 - 33*a^2*b^2 - 39*b^4 - 7*a*b*(a^2 + 63*b^2)*Sin[c + d*x]))/(9009
*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^6 - 165*a^4*b^2 + 450*a^2*b^4 + 195*b^6 - 24*a*b*(a^4
- 5*a^2*b^2 - 60*b^4)*Sin[c + d*x]))/(45045*b^4*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 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 2941

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

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 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {2}{15} \int \cos ^4(c+d x) \left (\frac {5 b}{2}+\frac {5}{2} a \sin (c+d x)\right ) (a+b \sin (c+d x))^{3/2} \, dx \\ & = -\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {4}{195} \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (20 a b+\frac {5}{4} \left (3 a^2+13 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {8 \int \frac {\cos ^4(c+d x) \left (\frac {5}{8} b \left (179 a^2+13 b^2\right )+\frac {15}{8} a \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{2145} \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {32 \int \frac {\cos ^2(c+d x) \left (-\frac {15}{16} b \left (a^4-474 a^2 b^2-39 b^4\right )-\frac {15}{2} a \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^2} \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac {128 \int \frac {\frac {15}{32} b \left (8 a^6-45 a^4 b^2+1890 a^2 b^4+195 b^6\right )+\frac {15}{32} a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{675675 b^4} \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac {\left (4 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{45045 b^5}-\frac {\left (4 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^5} \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac {\left (4 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{45045 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\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}{45045 b^5 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {8 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\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)}}{45045 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\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}}}{45045 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 16.06 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-256 a \left (32 a^7+32 a^6 b-189 a^5 b^2-189 a^4 b^3+570 a^3 b^4+570 a^2 b^5+1635 a b^6+1635 b^7\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+256 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (4096 a^7-23936 a^5 b^2-36512 a^3 b^4+67584 a b^6+8 \left (32 a^5 b^2-18192 a^3 b^4-18741 a b^6\right ) \cos (2 (c+d x))-224 \left (161 a^3 b^4-54 a b^6\right ) \cos (4 (c+d x))+20328 a b^6 \cos (6 (c+d x))+1024 a^6 b \sin (c+d x)-5840 a^4 b^3 \sin (c+d x)+186768 a^2 b^5 \sin (c+d x)+8151 b^7 \sin (c+d x)-80 a^4 b^3 \sin (3 (c+d x))-101688 a^2 b^5 \sin (3 (c+d x))-22269 b^7 \sin (3 (c+d x))-46536 a^2 b^5 \sin (5 (c+d x))-2457 b^7 \sin (5 (c+d x))+3003 b^7 \sin (7 (c+d x))\right )}{1441440 b^5 d \sqrt {a+b \sin (c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-256*a*(32*a^7 + 32*a^6*b - 189*a^5*b^2 - 189*a^4*b^3 + 570*a^3*b^4 + 570*a^2*b^5 + 1635*a*b^6 + 1635*b^7)*El
lipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 256*(32*a^8 - 197*a^6*b^2 +
615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*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]*(4096*a^7 - 23936*a^5*b^2 - 36512*a^3*b^4 + 67584*a*b^6 + 8*(32*a^5*b^2 - 18192*a^
3*b^4 - 18741*a*b^6)*Cos[2*(c + d*x)] - 224*(161*a^3*b^4 - 54*a*b^6)*Cos[4*(c + d*x)] + 20328*a*b^6*Cos[6*(c +
d*x)] + 1024*a^6*b*Sin[c + d*x] - 5840*a^4*b^3*Sin[c + d*x] + 186768*a^2*b^5*Sin[c + d*x] + 8151*b^7*Sin[c +
d*x] - 80*a^4*b^3*Sin[3*(c + d*x)] - 101688*a^2*b^5*Sin[3*(c + d*x)] - 22269*b^7*Sin[3*(c + d*x)] - 46536*a^2*
b^5*Sin[5*(c + d*x)] - 2457*b^7*Sin[5*(c + d*x)] + 3003*b^7*Sin[7*(c + d*x)]))/(1441440*b^5*d*Sqrt[a + b*Sin[c
+ d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1800\) vs. \(2(485)=970\).

Time = 38.12 (sec) , antiderivative size = 1801, normalized size of antiderivative = 3.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1801\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*(-780*((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)*E
llipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^9-128*((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^9-370*a^5*b^4+64*a^7*b^2+128*((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^8*b-96*((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^7*b^2-788*((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^3+576*((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^4+2460*((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^5+5280
*((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^6-1020*((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^
2*b^7-5760*((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)*Elli
pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^8+884*((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*b^2-3036*((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^4-4260*((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^6+6540*((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^8-780*a*b^8-3408*a^3*b^6+10164*
a*b^8*sin(d*x+c)^8+11634*a^2*b^7*sin(d*x+c)^7+4508*a^3*b^6*sin(d*x+c)^6-26922*a*b^8*sin(d*x+c)^6-5*a^4*b^5*sin
(d*x+c)^5-32532*a^2*b^7*sin(d*x+c)^5+8*a^5*b^4*sin(d*x+c)^4-13564*a^3*b^6*sin(d*x+c)^4+19302*a*b^8*sin(d*x+c)^
4-16*a^6*b^3*sin(d*x+c)^3+100*a^4*b^5*sin(d*x+c)^3+26382*a^2*b^7*sin(d*x+c)^3-64*a^7*b^2*sin(d*x+c)^2+362*a^5*
b^4*sin(d*x+c)^2+12464*a^3*b^6*sin(d*x+c)^2-1764*a*b^8*sin(d*x+c)^2+16*a^6*b^3*sin(d*x+c)-95*a^4*b^5*sin(d*x+c
)-5484*a^2*b^7*sin(d*x+c)+3003*b^9*sin(d*x+c)^9-7644*b^9*sin(d*x+c)^7+5109*b^9*sin(d*x+c)^5+312*b^9*sin(d*x+c)
^3-780*b^9*sin(d*x+c))/b^6/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 = 0.19 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.53 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\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 (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\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 ) + 6 \, \sqrt {2} {\left (32 i \, a^{7} b - 189 i \, a^{5} b^{3} + 570 i \, a^{3} b^{5} + 1635 i \, a b^{7}\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 ) + 6 \, \sqrt {2} {\left (-32 i \, a^{7} b + 189 i \, a^{5} b^{3} - 570 i \, a^{3} b^{5} - 1635 i \, a b^{7}\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 (3003 \, b^{8} \cos \left (d x + c\right )^{7} - 21 \, {\left (213 \, a^{2} b^{6} + 208 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 5 \, {\left (8 \, a^{4} b^{4} - 33 \, a^{2} b^{6} - 39 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (32 \, a^{6} b^{2} - 165 \, a^{4} b^{4} + 450 \, a^{2} b^{6} + 195 \, b^{8}\right )} \cos \left (d x + c\right ) - {\left (7161 \, a b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{3} b^{5} + 63 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (a^{5} b^{3} - 5 \, a^{3} b^{5} - 60 \, a b^{7}\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 )}}{135135 \, b^{6} d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/135135*(2*sqrt(2)*(64*a^8 - 402*a^6*b^2 + 1275*a^4*b^4 - 2400*a^2*b^6 - 585*b^8)*sqrt(I*b)*weierstrassPInve
rse(-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)*(64*a^8 - 402*a^6*b^2 + 1275*a^4*b^4 - 2400*a^2*b^6 - 585*b^8)*sqrt(-I*b)*weierstrassPInve
rse(-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) + 6*sqrt(2)*(32*I*a^7*b - 189*I*a^5*b^3 + 570*I*a^3*b^5 + 1635*I*a*b^7)*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)) + 6*sqrt(2)*(-32*I*a^7*b + 18
9*I*a^5*b^3 - 570*I*a^3*b^5 - 1635*I*a*b^7)*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*(3003*b^8*cos(d*x + c)^7 - 21*(213*a^2*b^6 + 208*b^8)*cos(d
*x + c)^5 - 5*(8*a^4*b^4 - 33*a^2*b^6 - 39*b^8)*cos(d*x + c)^3 + 2*(32*a^6*b^2 - 165*a^4*b^4 + 450*a^2*b^6 + 1
95*b^8)*cos(d*x + c) - (7161*a*b^7*cos(d*x + c)^5 - 35*(a^3*b^5 + 63*a*b^7)*cos(d*x + c)^3 + 48*(a^5*b^3 - 5*a
^3*b^5 - 60*a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d)

Sympy [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)*(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(5/2)*cos(d*x + c)^4*sin(d*x + c), x)

Giac [F]

\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^(5/2)*cos(d*x + c)^4*sin(d*x + c), x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int(cos(c + d*x)^4*sin(c + d*x)*(a + b*sin(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)^4*sin(c + d*x)*(a + b*sin(c + d*x))^(5/2), x)