3.8.43 \(\int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx\) [743]

Optimal. Leaf size=532 \[ \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^3 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/5*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(5/2)+8/15*(-a*d+b*c)^2*(2*a*c*d+b
*(c^2-3*d^2))*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(3/2)-2/15*(-a*d+b*c)*(a^2*d^2*(23*c^2+9*d^2)+2*a*
b*d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*d^4))*cos(f*x+e)/d^2/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(1/2)+2/15*(
-a*d+b*c)*(a^2*d^2*(23*c^2+9*d^2)+2*a*b*d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*d^4))*(sin(1/2*e+1/4*Pi+1/
2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*si
n(f*x+e))^(1/2)/d^3/(c^2-d^2)^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/15*(8*a^3*c*d^3-6*a*b^2*c*d*(c^2-5*d^2)-3*a
^2*b*d^2*(3*c^2+5*d^2)-b^3*(8*c^4-15*c^2*d^2+15*d^4))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2
*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/(c^2-d^2
)^2/f/(c+d*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.73, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2871, 3100, 2833, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) (b c-a d) \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) (b c-a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^3 f \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {8 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}+\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*(b*c -
a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(b*
c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[e + f
*x])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7
*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*
Sin[e + f*x]])/(15*d^3*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^
2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*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 2833

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

Rule 2871

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

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {2 \int \frac {\frac {1}{2} \left (2 b (b c-a d)^2-5 a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac {1}{2} \left (3 a (b c-a d)^2-5 b \left (a b c^2+\left (a^2+b^2\right ) c d-3 a b d^2\right )\right ) \sin (e+f x)-\frac {1}{2} b \left (2 a b c d-a^2 d^2+b^2 \left (4 c^2-5 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d \left (c^2-d^2\right )}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {4 \int \frac {-\frac {3}{4} d \left (24 a^2 b c d^2-a^3 d \left (5 c^2+3 d^2\right )-3 a b^2 d \left (3 c^2+5 d^2\right )-2 b^3 \left (c^3-5 c d^2\right )\right )-\frac {1}{4} \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 d^2 \left (c^2-d^2\right )^2}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {8 \int \frac {\frac {1}{8} d \left (3 a^2 b d^2 \left (27 c^2+5 d^2\right )-a^3 c d \left (15 c^2+17 d^2\right )-3 a b^2 d \left (7 c^3+25 c d^2\right )+b^3 \left (2 c^4+15 c^2 d^2+15 d^4\right )\right )+\frac {1}{8} (b c-a d) \left (8 b^2 c^4+14 a b c^3 d+23 a^2 c^2 d^2-21 b^2 c^2 d^2-78 a b c d^3+9 a^2 d^4+45 b^2 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d^2 \left (c^2-d^2\right )^3}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d^3 \left (c^2-d^2\right )^2}-\frac {\left ((b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{15 d^3 \left (c^2-d^2\right )^3}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {\left ((b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{15 d^3 \left (c^2-d^2\right )^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{15 d^3 \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^3 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.55, size = 584, normalized size = 1.10 \begin {gather*} \frac {2 \left (\frac {\left (d^2 \left (3 a^2 b d^2 \left (27 c^2+5 d^2\right )-a^3 c d \left (15 c^2+17 d^2\right )-3 a b^2 d \left (7 c^3+25 c d^2\right )+b^3 \left (2 c^4+15 c^2 d^2+15 d^4\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+\left (-a^3 d^3 \left (23 c^2+9 d^2\right )+3 a^2 b c d^2 \left (3 c^2+29 d^2\right )-3 a b^2 d \left (-2 c^4+19 c^2 d^2+15 d^4\right )+b^3 \left (8 c^5-21 c^3 d^2+45 c d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{5/2}}{(c-d)^3 (c+d)}+\frac {d (b c-a d) \cos (e+f x) \left (8 b^2 c^6+14 a b c^5 d+68 a^2 c^4 d^2-2 b^2 c^4 d^2-146 a b c^3 d^3+13 a^2 c^2 d^4+45 b^2 c^2 d^4-60 a b c d^5+15 a^2 d^6+45 b^2 d^6-d^2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (2 (e+f x))+2 d \left (2 a^2 c d^2 \left (27 c^2+5 d^2\right )+a b d \left (27 c^4-170 c^2 d^2+15 d^4\right )+b^2 \left (9 c^5-20 c^3 d^2+75 c d^4\right )\right ) \sin (e+f x)\right )}{2 \left (-c^2+d^2\right )^3}\right )}{15 d^3 f (c+d \sin (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(2*(((d^2*(3*a^2*b*d^2*(27*c^2 + 5*d^2) - a^3*c*d*(15*c^2 + 17*d^2) - 3*a*b^2*d*(7*c^3 + 25*c*d^2) + b^3*(2*c^
4 + 15*c^2*d^2 + 15*d^4))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (-(a^3*d^3*(23*c^2 + 9*d^2)) + 3*a
^2*b*c*d^2*(3*c^2 + 29*d^2) - 3*a*b^2*d*(-2*c^4 + 19*c^2*d^2 + 15*d^4) + b^3*(8*c^5 - 21*c^3*d^2 + 45*c*d^4))*
((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))
*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*(c + d)) + (d*(b*c - a*d)*Cos[e + f*x]*(8*b^2*c^6 + 14*a*b*c
^5*d + 68*a^2*c^4*d^2 - 2*b^2*c^4*d^2 - 146*a*b*c^3*d^3 + 13*a^2*c^2*d^4 + 45*b^2*c^2*d^4 - 60*a*b*c*d^5 + 15*
a^2*d^6 + 45*b^2*d^6 - d^2*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 +
45*d^4))*Cos[2*(e + f*x)] + 2*d*(2*a^2*c*d^2*(27*c^2 + 5*d^2) + a*b*d*(27*c^4 - 170*c^2*d^2 + 15*d^4) + b^2*(9
*c^5 - 20*c^3*d^2 + 75*c*d^4))*Sin[e + f*x]))/(2*(-c^2 + d^2)^3)))/(15*d^3*f*(c + d*Sin[e + f*x])^(5/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1620\) vs. \(2(574)=1148\).
time = 43.82, size = 1621, normalized size = 3.05

method	result	size
default	\(\text {Expression too large to display}\)	\(1621\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*b^3/d^3*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/
(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e
))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+3/d^3*b^2*(a*d-b*c)*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f
*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*
x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)
/(c+d))^(1/2))+2/(c^2-d^2)*d*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*
x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^
(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+1/d^3*(a^3*d^3-3*a^
2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(2/5/(c^2-d^2)/d^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)
^3+16/15*c/(c^2-d^2)^2/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)^2+2/15*d*cos(f*x+e)^2/(c^2
-d^2)^3*(23*c^2+9*d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2*d
^4-15*d^6)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(
1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/1
5*d*(23*c^2+9*d^2)/(c^2-d^2)^3*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(
f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d)
)^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+3*b/d^3*(a^2*d^2-
2*a*b*c*d+b^2*c^2)*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)^2+8/3*d*cos(f*x
+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(1/d*c-1)*((
c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c
)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*d*c/(c^2-d^2)^2*(1/d*c
-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*
x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+Elliptic
F(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

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((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 3078, normalized size = 5.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/45*((3*sqrt(2)*(16*b^3*c^7*d^2 + 12*a*b^2*c^6*d^3 + 6*(3*a^2*b - 8*b^3)*c^5*d^4 - (a^3 + 51*a*b^2)*c^4*d^5 -
3*(23*a^2*b - 15*b^3)*c^3*d^6 + 3*(11*a^3 + 45*a*b^2)*c^2*d^7 - 45*(a^2*b + b^3)*c*d^8)*cos(f*x + e)^2 + (sqr
t(2)*(16*b^3*c^6*d^3 + 12*a*b^2*c^5*d^4 + 6*(3*a^2*b - 8*b^3)*c^4*d^5 - (a^3 + 51*a*b^2)*c^3*d^6 - 3*(23*a^2*b
- 15*b^3)*c^2*d^7 + 3*(11*a^3 + 45*a*b^2)*c*d^8 - 45*(a^2*b + b^3)*d^9)*cos(f*x + e)^2 - sqrt(2)*(48*b^3*c^8*
d + 36*a*b^2*c^7*d^2 + 2*(27*a^2*b - 64*b^3)*c^6*d^3 - 3*(a^3 + 47*a*b^2)*c^5*d^4 - 3*(63*a^2*b - 29*b^3)*c^4*
d^5 + 2*(49*a^3 + 177*a*b^2)*c^3*d^6 - 6*(34*a^2*b + 15*b^3)*c^2*d^7 + 3*(11*a^3 + 45*a*b^2)*c*d^8 - 45*(a^2*b
+ b^3)*d^9))*sin(f*x + e) - sqrt(2)*(16*b^3*c^9 + 12*a*b^2*c^8*d + 18*a^2*b*c^7*d^2 - (a^3 + 15*a*b^2)*c^6*d^
3 - 3*(5*a^2*b + 33*b^3)*c^5*d^4 + 6*(5*a^3 - 3*a*b^2)*c^4*d^5 - 18*(14*a^2*b - 5*b^3)*c^3*d^6 + 9*(11*a^3 + 4
5*a*b^2)*c^2*d^7 - 135*(a^2*b + b^3)*c*d^8))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*
I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (3*sqrt(2)*(16*b^3*c^7*d^2 +
12*a*b^2*c^6*d^3 + 6*(3*a^2*b - 8*b^3)*c^5*d^4 - (a^3 + 51*a*b^2)*c^4*d^5 - 3*(23*a^2*b - 15*b^3)*c^3*d^6 + 3*
(11*a^3 + 45*a*b^2)*c^2*d^7 - 45*(a^2*b + b^3)*c*d^8)*cos(f*x + e)^2 + (sqrt(2)*(16*b^3*c^6*d^3 + 12*a*b^2*c^5
*d^4 + 6*(3*a^2*b - 8*b^3)*c^4*d^5 - (a^3 + 51*a*b^2)*c^3*d^6 - 3*(23*a^2*b - 15*b^3)*c^2*d^7 + 3*(11*a^3 + 45
*a*b^2)*c*d^8 - 45*(a^2*b + b^3)*d^9)*cos(f*x + e)^2 - sqrt(2)*(48*b^3*c^8*d + 36*a*b^2*c^7*d^2 + 2*(27*a^2*b
- 64*b^3)*c^6*d^3 - 3*(a^3 + 47*a*b^2)*c^5*d^4 - 3*(63*a^2*b - 29*b^3)*c^4*d^5 + 2*(49*a^3 + 177*a*b^2)*c^3*d^
6 - 6*(34*a^2*b + 15*b^3)*c^2*d^7 + 3*(11*a^3 + 45*a*b^2)*c*d^8 - 45*(a^2*b + b^3)*d^9))*sin(f*x + e) - sqrt(2
)*(16*b^3*c^9 + 12*a*b^2*c^8*d + 18*a^2*b*c^7*d^2 - (a^3 + 15*a*b^2)*c^6*d^3 - 3*(5*a^2*b + 33*b^3)*c^5*d^4 +
6*(5*a^3 - 3*a*b^2)*c^4*d^5 - 18*(14*a^2*b - 5*b^3)*c^3*d^6 + 9*(11*a^3 + 45*a*b^2)*c^2*d^7 - 135*(a^2*b + b^3
)*c*d^8))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*
cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(3*sqrt(2)*(8*I*b^3*c^6*d^3 + 6*I*a*b^2*c^5*d^4 + 3*I*(3*a^2
*b - 7*b^3)*c^4*d^5 - I*(23*a^3 + 57*a*b^2)*c^3*d^6 + 3*I*(29*a^2*b + 15*b^3)*c^2*d^7 - 9*I*(a^3 + 5*a*b^2)*c*
d^8)*cos(f*x + e)^2 + (sqrt(2)*(8*I*b^3*c^5*d^4 + 6*I*a*b^2*c^4*d^5 + 3*I*(3*a^2*b - 7*b^3)*c^3*d^6 - I*(23*a^
3 + 57*a*b^2)*c^2*d^7 + 3*I*(29*a^2*b + 15*b^3)*c*d^8 - 9*I*(a^3 + 5*a*b^2)*d^9)*cos(f*x + e)^2 + sqrt(2)*(-24
*I*b^3*c^7*d^2 - 18*I*a*b^2*c^6*d^3 - I*(27*a^2*b - 55*b^3)*c^5*d^4 + 3*I*(23*a^3 + 55*a*b^2)*c^4*d^5 - 6*I*(4
5*a^2*b + 19*b^3)*c^3*d^6 + 2*I*(25*a^3 + 96*a*b^2)*c^2*d^7 - 3*I*(29*a^2*b + 15*b^3)*c*d^8 + 9*I*(a^3 + 5*a*b
^2)*d^9))*sin(f*x + e) + sqrt(2)*(-8*I*b^3*c^8*d - 6*I*a*b^2*c^7*d^2 - 3*I*(3*a^2*b + b^3)*c^6*d^3 + I*(23*a^3
+ 39*a*b^2)*c^5*d^4 - 6*I*(19*a^2*b - 3*b^3)*c^4*d^5 + 6*I*(13*a^3 + 36*a*b^2)*c^3*d^6 - 9*I*(29*a^2*b + 15*b
^3)*c^2*d^7 + 27*I*(a^3 + 5*a*b^2)*c*d^8))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3
- 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(
f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(3*sqrt(2)*(-8*I*b^3*c^6*d^3 - 6*I*a*b^2*c^5*d^4 - 3*I*(3*a^2*b
- 7*b^3)*c^4*d^5 + I*(23*a^3 + 57*a*b^2)*c^3*d^6 - 3*I*(29*a^2*b + 15*b^3)*c^2*d^7 + 9*I*(a^3 + 5*a*b^2)*c*d^
8)*cos(f*x + e)^2 + (sqrt(2)*(-8*I*b^3*c^5*d^4 - 6*I*a*b^2*c^4*d^5 - 3*I*(3*a^2*b - 7*b^3)*c^3*d^6 + I*(23*a^3
+ 57*a*b^2)*c^2*d^7 - 3*I*(29*a^2*b + 15*b^3)*c*d^8 + 9*I*(a^3 + 5*a*b^2)*d^9)*cos(f*x + e)^2 + sqrt(2)*(24*I
*b^3*c^7*d^2 + 18*I*a*b^2*c^6*d^3 + I*(27*a^2*b - 55*b^3)*c^5*d^4 - 3*I*(23*a^3 + 55*a*b^2)*c^4*d^5 + 6*I*(45*
a^2*b + 19*b^3)*c^3*d^6 - 2*I*(25*a^3 + 96*a*b^2)*c^2*d^7 + 3*I*(29*a^2*b + 15*b^3)*c*d^8 - 9*I*(a^3 + 5*a*b^2
)*d^9))*sin(f*x + e) + sqrt(2)*(8*I*b^3*c^8*d + 6*I*a*b^2*c^7*d^2 + 3*I*(3*a^2*b + b^3)*c^6*d^3 - I*(23*a^3 +
39*a*b^2)*c^5*d^4 + 6*I*(19*a^2*b - 3*b^3)*c^4*d^5 - 6*I*(13*a^3 + 36*a*b^2)*c^3*d^6 + 9*I*(29*a^2*b + 15*b^3)
*c^2*d^7 - 27*I*(a^3 + 5*a*b^2)*c*d^8))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 +
9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(
f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 6*((8*b^3*c^5*d^4 + 6*a*b^2*c^4*d^5 + 3*(3*a^2*b - 7*b^3)*c^3*d^6
- (23*a^3 + 57*a*b^2)*c^2*d^7 + 3*(29*a^2*b + 15*b^3)*c*d^8 - 9*(a^3 + 5*a*b^2)*d^9)*cos(f*x + e)^3 - (9*b^3*
c^6*d^3 + 18*a*b^2*c^5*d^4 - 15*a^2*b*d^9 + (27*a^2*b - 20*b^3)*c^4*d^5 - 6*(9*a^3 + 25*a*b^2)*c^3*d^6 + 15*(1
2*a^2*b + 5*b^3)*c^2*d^7 - 10*(a^3 + 6*a*b^2)*c*d^8)*cos(f*x + e)*sin(f*x + e) - (4*b^3*c^7*d^2 + 3*a*b^2*c^6*
d^3 + 3*(9*a^2*b + b^3)*c^5*d^4 - (34*a^3 + 69*a*b^2)*c^4*d^5 + 12*(7*a^2*b + b^3)*c^3*d^6 - 9*(2*a^3 + 9*a*b^
2)*c^2*d^7 + 9*(9*a^2*b + 5*b^3)*c*d^8 - 3*(4*a^3 + 15*a*b^2)*d^9)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(3*
(c^7*d^6 - 3*c^5*d^8 + 3*c^3*d^10 - c*d^12)*f*c...

________________________________________________________________________________________

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((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2),x)

[Out]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2), x)

________________________________________________________________________________________