\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [1184]

Optimal result

Integrand size = 31, antiderivative size = 411 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {16 a \left (32 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-2/3*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c))^(3/2)+2/3*(11*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)
^3/a^2/b^2/d/(a+b*sin(d*x+c))^(1/2)-8/21*(32*a^2-11*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+8/21*(24*a^2-
7*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a/b^4/d-2/21*(80*a^2-21*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*
sin(d*x+c))^(1/2)/a^2/b^3/d+16/21*a*(32*a^2-15*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)-8/21*(64*a^4-46*a^2*b^2+3*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)*El
lipticF(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)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2970, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac {16 a \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}+\frac {8 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{21 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(11*a^2 - 3*b^2)*Cos[
c + d*x]*Sin[c + d*x]^3)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(32*a^2 - 11*b^2)*Cos[c + d*x]*Sqrt[a + b
*Sin[c + d*x]])/(21*b^5*d) + (8*(24*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(21*a*b^4
*d) - (2*(80*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(21*a^2*b^3*d) - (16*a*(32*a^
2 - 15*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)]) + (8*(64*a^4 - 46*a^2*b^2 + 3*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]])

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 2970

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f
*(m + 1))), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b
*Sin[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2))), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n +
4, 0])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \int \frac {\sin ^2(c+d x) \left (\frac {15}{4} \left (4 a^2-b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^2 b^2} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {8 \int \frac {\sin (c+d x) \left (-\frac {1}{2} a \left (80 a^2-21 b^2\right )+\frac {5}{2} a^2 b \sin (c+d x)+\frac {5}{2} a \left (24 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 a^2 b^3} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {16 \int \frac {\frac {5}{2} a^2 \left (24 a^2-7 b^2\right )-10 a^3 b \sin (c+d x)-\frac {15}{4} a^2 \left (32 a^2-11 b^2\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 a^2 b^4} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {32 \int \frac {\frac {15}{8} a^2 b \left (16 a^2-3 b^2\right )+\frac {15}{4} a^3 \left (32 a^2-15 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 a^2 b^5} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {\left (8 a \left (32 a^2-15 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{21 b^6}+\frac {\left (4 \left (64 a^4-46 a^2 b^2+3 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^6} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {\left (8 a \left (32 a^2-15 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 (4 \left (64 a^4-46 a^2 b^2+3 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 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {16 a \left (32 a^2-15 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 {8 \left (64 a^4-46 a^2 b^2+3 b^4\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}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.96 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {32 a (a+b)^2 \left (32 a^2-15 b^2\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}-16 (a+b) \left (64 a^4-46 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}-\frac {1}{2} b \cos (c+d x) \left (1024 a^4-288 a^2 b^2-27 b^4-8 \left (8 a^2 b^2-3 b^4\right ) \cos (2 (c+d x))+3 b^4 \cos (4 (c+d x))+1280 a^3 b \sin (c+d x)-516 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))\right )}{42 b^6 d (a+b \sin (c+d x))^{3/2}} \]

[In]

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

[Out]

(32*a*(a + b)^2*(32*a^2 - 15*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b
))^(3/2) - 16*(a + b)*(64*a^4 - 46*a^2*b^2 + 3*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Si
n[c + d*x])/(a + b))^(3/2) - (b*Cos[c + d*x]*(1024*a^4 - 288*a^2*b^2 - 27*b^4 - 8*(8*a^2*b^2 - 3*b^4)*Cos[2*(c
+ d*x)] + 3*b^4*Cos[4*(c + d*x)] + 1280*a^3*b*Sin[c + d*x] - 516*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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1641\) vs. \(2(445)=890\).

Time = 1.94 (sec) , antiderivative size = 1642, normalized size of antiderivative = 4.00

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

[In]

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

[Out]

-2/21*(3*b^6*cos(d*x+c)^6+6*a*b^5*cos(d*x+c)^4*sin(d*x+c)+(-16*a^2*b^4+3*b^6)*cos(d*x+c)^4+(160*a^3*b^3-66*a*b
^5)*cos(d*x+c)^2*sin(d*x+c)+(128*a^4*b^2-28*a^2*b^4-6*b^6)*cos(d*x+c)^2+4*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*
(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*b*(64*EllipticF((b/(a-b)*sin(d*x+c)+a/(
a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b-48*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^
3*b^2-46*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^3+27*EllipticF((b/(a-b)*sin(d
*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4+3*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(
1/2))*b^5-64*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5+94*EllipticE((b/(a-b)*sin(d
*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2-30*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b)
)^(1/2))*a*b^4)*sin(d*x+c)+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)*Ellipti
cF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(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)*sin(d*x+c)+a/(a-b))^(1/2)
,((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^4*b^2-184*(-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)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(
d*x+c)+a/(a-b))^(1/2)*a^3*b^3+108*(-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)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^2*b^4+12*(-
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)+a/(a-b))^(
1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a*b^5-256*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b
/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*si
n(d*x+c)+a/(a-b))^(1/2)*a^6+376*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*Ellipt
icE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^4*b^2-120*(-b
/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1
/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^2*b^4)/(a+b*sin(d*x+c))^(3/2)/b^7/cos(d*x+c)/d

Fricas [C] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 873, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/63*(2*(sqrt(2)*(128*a^4*b^2 - 108*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(128*a^5*b - 108*a^3*b^3 + 9*a
*b^5)*sin(d*x + c) - sqrt(2)*(128*a^6 + 20*a^4*b^2 - 99*a^2*b^4 + 9*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)*(128*a^4*b^2 - 108*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(128*a^5*b - 108*a^3*b^3 + 9*a*b^5)*
sin(d*x + c) - sqrt(2)*(128*a^6 + 20*a^4*b^2 - 99*a^2*b^4 + 9*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*(
sqrt(2)*(32*I*a^3*b^3 - 15*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)*(-32*I*a^4*b^2 + 15*I*a^2*b^4)*sin(d*x + c) + s
qrt(2)*(-32*I*a^5*b - 17*I*a^3*b^3 + 15*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^3*b^3 + 15*I*a*b^5)*cos(d*x + c)^2 +
2*sqrt(2)*(32*I*a^4*b^2 - 15*I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(32*I*a^5*b + 17*I*a^3*b^3 - 15*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 - (16*a^2*b^4 - 3*b^6)*cos(d*x + c)^3 + 2*(64*a^4*b^2 - 14*a^2*b^4 - 3*b^6)*cos(d*x +
c) + 2*(3*a*b^5*cos(d*x + c)^3 + (80*a^3*b^3 - 33*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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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