\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx\) [327]

Optimal result

Integrand size = 37, antiderivative size = 519 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]

[Out]

1/4*d^(3/2)*(3*A*d*(21*c^2+30*c*d+13*d^2)-B*(35*c^3+70*c^2*d+67*c*d^2+20*d^3))*arctanh(cos(f*x+e)*a^(1/2)*d^(1
/2)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^5/(c+d)^(5/2)/f-1/4*(A-B)*cos(f*x+e)/(c-d)/f/(a+a*sin(f*
x+e))^(5/2)/(c+d*sin(f*x+e))^2-1/16*(3*A*c-19*A*d+5*B*c+11*B*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/
(c+d*sin(f*x+e))^2-1/32*(B*(5*c^2-82*c*d-115*d^2)+3*A*(c^2-10*c*d+73*d^2))*arctanh(1/2*cos(f*x+e)*a^(1/2)*2^(1
/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^5/f*2^(1/2)-1/16*d*(A*(3*c^2-20*c*d-31*d^2)+B*(5*c^2+28*c*d+15*d^2))
*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2)-1/16*d*(3*A*(c^3-7*c^2*d-37*c*d^2-21
*d^3)+B*(5*c^3+73*c^2*d+79*c*d^2+35*d^3))*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(
1/2)

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3057, 3063, 3064, 2728, 212, 2852, 214} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {\left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^5}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {(3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2} \]

[In]

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

-1/16*((B*(5*c^2 - 82*c*d - 115*d^2) + 3*A*(c^2 - 10*c*d + 73*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sq
rt[a + a*Sin[e + f*x]])])/(Sqrt[2]*a^(5/2)*(c - d)^5*f) + (d^(3/2)*(3*A*d*(21*c^2 + 30*c*d + 13*d^2) - B*(35*c
^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x
]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - ((A - B)*Cos[e + f*x])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(
c + d*Sin[e + f*x])^2) - ((3*A*c + 5*B*c - 19*A*d + 11*B*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x
])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(A*(3*c^2 - 20*c*d - 31*d^2) + B*(5*c^2 + 28*c*d + 15*d^2))*Cos[e + f*x]
)/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (d*(3*A*(c^3 - 7*c^2*d - 37*c
*d^2 - 21*d^3) + B*(5*c^3 + 73*c^2*d + 79*c*d^2 + 35*d^3))*Cos[e + f*x])/(16*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a
+ a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2852

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

Rule 3057

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

Rule 3063

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

Rule 3064

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(
B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a (3 A c+5 B c-12 A d+4 B d)-\frac {7}{2} a (A-B) d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx}{4 a^2 (c-d)} \\ & = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac {\int \frac {\frac {1}{4} a^2 \left (B \left (5 c^2-57 c d-60 d^2\right )+A \left (3 c^2-15 c d+124 d^2\right )\right )+\frac {5}{4} a^2 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx}{8 a^4 (c-d)^2} \\ & = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a^3 \left (B \left (5 c^3-62 c^2 d-113 c d^2-70 d^3\right )+3 A \left (c^3-6 c^2 d+43 c d^2+42 d^3\right )\right )-\frac {3}{2} a^3 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{16 a^5 (c-d)^3 (c+d)} \\ & = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {\int \frac {\frac {1}{2} a^4 \left (B \left (5 c^4-67 c^3 d-201 c^2 d^2-233 c d^3-80 d^4\right )+3 A \left (c^4-7 c^3 d+47 c^2 d^2+99 c d^3+52 d^4\right )\right )+\frac {1}{2} a^4 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{16 a^6 (c-d)^4 (c+d)^2} \\ & = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2 (c-d)^5}-\frac {\left (d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 a^3 (c-d)^5 (c+d)^2} \\ & = -\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{16 a^2 (c-d)^5 f}+\frac {\left (d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{4 a^2 (c-d)^5 (c+d)^2 f} \\ & = -\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 16.01 (sec) , antiderivative size = 2465, normalized size of antiderivative = 4.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Result too large to show} \]

[In]

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

((1 + I)*(3*A*c^2 + 5*B*c^2 - 30*A*c*d - 82*B*c*d + 219*A*d^2 - 115*B*d^2)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[
(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((16*(-1)^(1/4)*c
^5 - 80*(-1)^(1/4)*c^4*d + 160*(-1)^(1/4)*c^3*d^2 - 160*(-1)^(1/4)*c^2*d^3 + 80*(-1)^(1/4)*c*d^4 - 16*(-1)^(1/
4)*d^5)*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d +
67*c*d^2 + 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4
& , (-(d*Log[-#1 + Tan[(e + f*x)/4]]) + Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 + Tan[(e +
f*x)/4]]*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - S
qrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1^
2 - c*#1^3) & ])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))
^(5/2)) + (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*(e + f*x -
2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4 & , (-(d*Log[-#1 + Tan[(e + f*x)
/4]]) - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 + Tan[(e + f*x)/4]]*#1 - 2*Sqrt[d]*Sqrt[c
+ d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 + Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan
[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1^2 - c*#1^3) & ])*(Cos[(e + f*x)/
2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[(e + f*x)/2] + S
in[(e + f*x)/2])*(-44*A*c^5*Cos[(e + f*x)/2] + 12*B*c^5*Cos[(e + f*x)/2] + 84*A*c^4*d*Cos[(e + f*x)/2] - 116*B
*c^4*d*Cos[(e + f*x)/2] + 249*A*c^3*d^2*Cos[(e + f*x)/2] - 433*B*c^3*d^2*Cos[(e + f*x)/2] + 385*A*c^2*d^3*Cos[
(e + f*x)/2] - 277*B*c^2*d^3*Cos[(e + f*x)/2] + 239*A*c*d^4*Cos[(e + f*x)/2] - 95*B*c*d^4*Cos[(e + f*x)/2] + 4
7*A*d^5*Cos[(e + f*x)/2] - 51*B*d^5*Cos[(e + f*x)/2] - 12*A*c^5*Cos[(3*(e + f*x))/2] - 20*B*c^5*Cos[(3*(e + f*
x))/2] + 40*A*c^4*d*Cos[(3*(e + f*x))/2] - 104*B*c^4*d*Cos[(3*(e + f*x))/2] + 261*A*c^3*d^2*Cos[(3*(e + f*x))/
2] - 581*B*c^3*d^2*Cos[(3*(e + f*x))/2] + 781*A*c^2*d^3*Cos[(3*(e + f*x))/2] - 665*B*c^2*d^3*Cos[(3*(e + f*x))
/2] + 579*A*c*d^4*Cos[(3*(e + f*x))/2] - 299*B*c*d^4*Cos[(3*(e + f*x))/2] + 79*A*d^5*Cos[(3*(e + f*x))/2] - 59
*B*d^5*Cos[(3*(e + f*x))/2] + 12*A*c^4*d*Cos[(5*(e + f*x))/2] + 20*B*c^4*d*Cos[(5*(e + f*x))/2] - 73*A*c^3*d^2
*Cos[(5*(e + f*x))/2] + 217*B*c^3*d^2*Cos[(5*(e + f*x))/2] - 353*A*c^2*d^3*Cos[(5*(e + f*x))/2] + 397*B*c^2*d^
3*Cos[(5*(e + f*x))/2] - 419*A*c*d^4*Cos[(5*(e + f*x))/2] + 251*B*c*d^4*Cos[(5*(e + f*x))/2] - 127*A*d^5*Cos[(
5*(e + f*x))/2] + 75*B*d^5*Cos[(5*(e + f*x))/2] + 3*A*c^3*d^2*Cos[(7*(e + f*x))/2] + 5*B*c^3*d^2*Cos[(7*(e + f
*x))/2] - 21*A*c^2*d^3*Cos[(7*(e + f*x))/2] + 73*B*c^2*d^3*Cos[(7*(e + f*x))/2] - 111*A*c*d^4*Cos[(7*(e + f*x)
)/2] + 79*B*c*d^4*Cos[(7*(e + f*x))/2] - 63*A*d^5*Cos[(7*(e + f*x))/2] + 35*B*d^5*Cos[(7*(e + f*x))/2] + 44*A*
c^5*Sin[(e + f*x)/2] - 12*B*c^5*Sin[(e + f*x)/2] - 84*A*c^4*d*Sin[(e + f*x)/2] + 116*B*c^4*d*Sin[(e + f*x)/2]
- 249*A*c^3*d^2*Sin[(e + f*x)/2] + 433*B*c^3*d^2*Sin[(e + f*x)/2] - 385*A*c^2*d^3*Sin[(e + f*x)/2] + 277*B*c^2
*d^3*Sin[(e + f*x)/2] - 239*A*c*d^4*Sin[(e + f*x)/2] + 95*B*c*d^4*Sin[(e + f*x)/2] - 47*A*d^5*Sin[(e + f*x)/2]
+ 51*B*d^5*Sin[(e + f*x)/2] - 12*A*c^5*Sin[(3*(e + f*x))/2] - 20*B*c^5*Sin[(3*(e + f*x))/2] + 40*A*c^4*d*Sin[
(3*(e + f*x))/2] - 104*B*c^4*d*Sin[(3*(e + f*x))/2] + 261*A*c^3*d^2*Sin[(3*(e + f*x))/2] - 581*B*c^3*d^2*Sin[(
3*(e + f*x))/2] + 781*A*c^2*d^3*Sin[(3*(e + f*x))/2] - 665*B*c^2*d^3*Sin[(3*(e + f*x))/2] + 579*A*c*d^4*Sin[(3
*(e + f*x))/2] - 299*B*c*d^4*Sin[(3*(e + f*x))/2] + 79*A*d^5*Sin[(3*(e + f*x))/2] - 59*B*d^5*Sin[(3*(e + f*x))
/2] - 12*A*c^4*d*Sin[(5*(e + f*x))/2] - 20*B*c^4*d*Sin[(5*(e + f*x))/2] + 73*A*c^3*d^2*Sin[(5*(e + f*x))/2] -
217*B*c^3*d^2*Sin[(5*(e + f*x))/2] + 353*A*c^2*d^3*Sin[(5*(e + f*x))/2] - 397*B*c^2*d^3*Sin[(5*(e + f*x))/2] +
419*A*c*d^4*Sin[(5*(e + f*x))/2] - 251*B*c*d^4*Sin[(5*(e + f*x))/2] + 127*A*d^5*Sin[(5*(e + f*x))/2] - 75*B*d
^5*Sin[(5*(e + f*x))/2] + 3*A*c^3*d^2*Sin[(7*(e + f*x))/2] + 5*B*c^3*d^2*Sin[(7*(e + f*x))/2] - 21*A*c^2*d^3*S
in[(7*(e + f*x))/2] + 73*B*c^2*d^3*Sin[(7*(e + f*x))/2] - 111*A*c*d^4*Sin[(7*(e + f*x))/2] + 79*B*c*d^4*Sin[(7
*(e + f*x))/2] - 63*A*d^5*Sin[(7*(e + f*x))/2] + 35*B*d^5*Sin[(7*(e + f*x))/2]))/(128*(c - d)^4*(c + d)^2*f*(a
*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7321\) vs. \(2(476)=952\).

Time = 3.45 (sec) , antiderivative size = 7322, normalized size of antiderivative = 14.11

method	result	size
default	\(\text {Expression too large to display}\)	\(7322\)

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4135 vs. \(2 (476) = 952\).

Time = 24.73 (sec) , antiderivative size = 8555, normalized size of antiderivative = 16.48 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Timed out

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2024 vs. \(2 (476) = 952\).

Time = 1.00 (sec) , antiderivative size = 2024, normalized size of antiderivative = 3.90 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-1/32*(4*sqrt(2)*(35*sqrt(2)*B*sqrt(a)*c^3*d^2 - 63*sqrt(2)*A*sqrt(a)*c^2*d^3 + 70*sqrt(2)*B*sqrt(a)*c^2*d^3 -
90*sqrt(2)*A*sqrt(a)*c*d^4 + 67*sqrt(2)*B*sqrt(a)*c*d^4 - 39*sqrt(2)*A*sqrt(a)*d^5 + 20*sqrt(2)*B*sqrt(a)*d^5
)*arctan(sqrt(2)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)/sqrt(-c*d - d^2))/((a^3*c^7*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*
e)) - 3*a^3*c^6*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a^3*c^5*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^
3*c^4*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*a^3*c^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^3*c^2*d^
5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*a^3*c*d^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^3*d^7*sgn(cos(-1/4
*pi + 1/2*f*x + 1/2*e)))*sqrt(-c*d - d^2)) - (3*A*sqrt(a)*c^2 + 5*B*sqrt(a)*c^2 - 30*A*sqrt(a)*c*d - 82*B*sqrt
(a)*c*d + 219*A*sqrt(a)*d^2 - 115*B*sqrt(a)*d^2)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sqrt(2)*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*sqrt(2)*a^3*c^3
*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 10*sqrt(2)*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt
(2)*a^3*c*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + (3*
A*sqrt(a)*c^2 + 5*B*sqrt(a)*c^2 - 30*A*sqrt(a)*c*d - 82*B*sqrt(a)*c*d + 219*A*sqrt(a)*d^2 - 115*B*sqrt(a)*d^2)
*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sqrt(2)*a^3
*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*sqrt(2)*a^3*c^3*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 10*s
qrt(2)*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt(2)*a^3*c*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e
)) - sqrt(2)*a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + 2*(12*A*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/
2*e)^7 + 20*B*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 84*A*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x +
1/2*e)^7 + 292*B*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 444*A*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)^7 + 316*B*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 252*A*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x +
1/2*e)^7 + 140*B*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 12*A*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2
*e)^5 - 20*B*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 52*A*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2
*e)^5 - 252*B*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 500*A*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x +
1/2*e)^5 - 908*B*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 1196*A*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*
x + 1/2*e)^5 - 804*B*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 568*A*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)^5 - 320*B*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 3*A*sqrt(a)*c^5*sin(-1/4*pi + 1/2*f*x + 1/2*
e)^3 + 5*B*sqrt(a)*c^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 5*A*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 +
51*B*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 146*A*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3
+ 434*B*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 710*A*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e
)^3 + 918*B*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 1057*A*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/
2*e)^3 + 665*B*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 399*A*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*
e)^3 + 231*B*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 5*A*sqrt(a)*c^5*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3
*B*sqrt(a)*c^5*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 9*A*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 33*B*sqrt(a
)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 86*A*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 206*B*sqrt(a)*c
^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 290*A*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 298*B*sqrt(a)*c
^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 303*A*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 175*B*sqrt(a)*c*d
^4*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 85*A*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 53*B*sqrt(a)*d^5*sin(-1/
4*pi + 1/2*f*x + 1/2*e))/((sqrt(2)*a^3*c^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*sqrt(2)*a^3*c^5*d*sgn(cos(-
1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*c^4*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 4*sqrt(2)*a^3*c^3*d^3*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*c^2*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*sqrt(2)*a^3*c
*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*(2*d*sin(-1/4*
pi + 1/2*f*x + 1/2*e)^4 - c*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 3*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 + c + d)^2
))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]

[In]

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3),x)

[Out]

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3), x)