3.1160 \(\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

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

[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)

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Rubi [A]  time = 1.07, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2+8 a^4-39 b^4\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (-24 a b \left (-5 a^2 b^2+a^4-60 b^4\right ) \sin (c+d x)-165 a^4 b^2+450 a^2 b^4+32 a^6+195 b^6\right )}{45045 b^4 d}-\frac {8 \left (-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (-189 a^4 b^2+570 a^2 b^4+32 a^6+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d} \]

Antiderivative was successfully verified.

[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 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]

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 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 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 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 2862

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] && Gt
Q[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 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]

Rubi steps

\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\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 ) 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}}}{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*}

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Mathematica [A]  time = 21.65, size = 450, normalized size = 1.00 \[ \frac {256 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+b \cos (c+d x) \left (4096 a^7+1024 a^6 b \sin (c+d x)-23936 a^5 b^2-5840 a^4 b^3 \sin (c+d x)-80 a^4 b^3 \sin (3 (c+d x))-36512 a^3 b^4-224 \left (161 a^3 b^4-54 a b^6\right ) \cos (4 (c+d x))+186768 a^2 b^5 \sin (c+d x)-101688 a^2 b^5 \sin (3 (c+d x))-46536 a^2 b^5 \sin (5 (c+d x))+8 \left (32 a^5 b^2-18192 a^3 b^4-18741 a b^6\right ) \cos (2 (c+d x))+20328 a b^6 \cos (6 (c+d x))+67584 a b^6+8151 b^7 \sin (c+d x)-22269 b^7 \sin (3 (c+d x))-2457 b^7 \sin (5 (c+d x))+3003 b^7 \sin (7 (c+d x))\right )-256 a \left (32 a^7+32 a^6 b-189 a^5 b^2-189 a^4 b^3+570 a^3 b^4+570 a^2 b^5+1635 a b^6+1635 b^7\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{1441440 b^5 d \sqrt {a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[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]])

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fricas [F]  time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (2 \, a b \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} + {\left (b^{2} \cos \left (d x + c\right )^{6} - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(2*a*b*cos(d*x + c)^6 - 2*a*b*cos(d*x + c)^4 + (b^2*cos(d*x + c)^6 - (a^2 + b^2)*cos(d*x + c)^4)*sin
(d*x + c))*sqrt(b*sin(d*x + c) + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 1.86, size = 1801, normalized size = 3.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(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)+10164*a*b^8
*sin(d*x+c)^8-780*a*b^8+64*a^7*b^2-3408*a^3*b^6-370*a^5*b^4+11634*a^2*b^7*sin(d*x+c)^7-128*((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^9-788*((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+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)+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^4-5760*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s
in(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^8+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+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
^2-780*((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)*Elliptic
F(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^9-3036*((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)*El
lipticE(((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+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)/b^6/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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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