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

Optimal. Leaf size=519 \[ -\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]

[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)

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Rubi [A]
time = 1.46, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3057, 3063, 3064, 2728, 212, 2852, 214} \begin {gather*} -\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.05, size = 2103, normalized size = 4.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Co
s[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(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] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e
+ f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(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] + Sin[(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] + 47*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*C
os[(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*Sin[(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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7321\) vs. \(2(476)=952\).
time = 32.62, size = 7322, normalized size = 14.11

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4188 vs. \(2 (492) = 984\).
time = 27.88, size = 8675, normalized size = 16.71 \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))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/64*(sqrt(2)*(4*(3*A + 5*B)*c^6 - 8*(9*A + 31*B)*c^5*d + 4*(117*A - 413*B)*c^4*d^2 + 16*(177*A - 233*B)*c^3*
d^3 + 4*(1197*A - 1013*B)*c^2*d^4 + 8*(423*A - 271*B)*c*d^5 + 4*(219*A - 115*B)*d^6 + ((3*A + 5*B)*c^4*d^2 - 2
4*(A + 3*B)*c^3*d^3 + 2*(81*A - 137*B)*c^2*d^4 + 24*(17*A - 13*B)*c*d^5 + (219*A - 115*B)*d^6)*cos(f*x + e)^5
+ (2*(3*A + 5*B)*c^5*d - 3*(13*A + 43*B)*c^4*d^2 + 4*(63*A - 191*B)*c^3*d^3 + 6*(217*A - 241*B)*c^2*d^4 + 2*(8
31*A - 583*B)*c*d^5 + 3*(219*A - 115*B)*d^6)*cos(f*x + e)^4 - ((3*A + 5*B)*c^6 - 4*(3*A + 13*B)*c^5*d + (75*A
- 547*B)*c^4*d^2 + 8*(123*A - 203*B)*c^3*d^3 + 19*(123*A - 115*B)*c^2*d^4 + 4*(525*A - 349*B)*c*d^5 + 3*(219*A
- 115*B)*d^6)*cos(f*x + e)^3 - (3*(3*A + 5*B)*c^6 - 2*(21*A + 83*B)*c^5*d + (267*A - 1507*B)*c^4*d^2 + 4*(669
*A - 1045*B)*c^3*d^3 + (5871*A - 5383*B)*c^2*d^4 + 2*(2523*A - 1667*B)*c*d^5 + 7*(219*A - 115*B)*d^6)*cos(f*x
+ e)^2 + 2*((3*A + 5*B)*c^6 - 2*(9*A + 31*B)*c^5*d + (117*A - 413*B)*c^4*d^2 + 4*(177*A - 233*B)*c^3*d^3 + (11
97*A - 1013*B)*c^2*d^4 + 2*(423*A - 271*B)*c*d^5 + (219*A - 115*B)*d^6)*cos(f*x + e) + (4*(3*A + 5*B)*c^6 - 8*
(9*A + 31*B)*c^5*d + 4*(117*A - 413*B)*c^4*d^2 + 16*(177*A - 233*B)*c^3*d^3 + 4*(1197*A - 1013*B)*c^2*d^4 + 8*
(423*A - 271*B)*c*d^5 + 4*(219*A - 115*B)*d^6 + ((3*A + 5*B)*c^4*d^2 - 24*(A + 3*B)*c^3*d^3 + 2*(81*A - 137*B)
*c^2*d^4 + 24*(17*A - 13*B)*c*d^5 + (219*A - 115*B)*d^6)*cos(f*x + e)^4 - 2*((3*A + 5*B)*c^5*d - (21*A + 67*B)
*c^4*d^2 + 2*(69*A - 173*B)*c^3*d^3 + 2*(285*A - 293*B)*c^2*d^4 + (627*A - 427*B)*c*d^5 + (219*A - 115*B)*d^6)
*cos(f*x + e)^3 - ((3*A + 5*B)*c^6 - 6*(A + 7*B)*c^5*d + 3*(11*A - 227*B)*c^4*d^2 + 12*(105*A - 193*B)*c^3*d^3
+ 3*(1159*A - 1119*B)*c^2*d^4 + 6*(559*A - 375*B)*c*d^5 + 5*(219*A - 115*B)*d^6)*cos(f*x + e)^2 + 2*((3*A + 5
*B)*c^6 - 2*(9*A + 31*B)*c^5*d + (117*A - 413*B)*c^4*d^2 + 4*(177*A - 233*B)*c^3*d^3 + (1197*A - 1013*B)*c^2*d
^4 + 2*(423*A - 271*B)*c*d^5 + (219*A - 115*B)*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2
- 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*
x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(
140*B*a*c^5*d - 28*(9*A - 20*B)*a*c^4*d^2 - 8*(108*A - 121*B)*a*c^3*d^3 - 8*(141*A - 112*B)*a*c^2*d^4 - 4*(168
*A - 107*B)*a*c*d^5 - 4*(39*A - 20*B)*a*d^6 + (35*B*a*c^3*d^3 - 7*(9*A - 10*B)*a*c^2*d^4 - (90*A - 67*B)*a*c*d
^5 - (39*A - 20*B)*a*d^6)*cos(f*x + e)^5 + (70*B*a*c^4*d^2 - 7*(18*A - 35*B)*a*c^3*d^3 - (369*A - 344*B)*a*c^2
*d^4 - (348*A - 241*B)*a*c*d^5 - 3*(39*A - 20*B)*a*d^6)*cos(f*x + e)^4 - (35*B*a*c^5*d - 21*(3*A - 10*B)*a*c^4
*d^2 - 2*(171*A - 226*B)*a*c^3*d^3 - 6*(98*A - 83*B)*a*c^2*d^4 - (426*A - 281*B)*a*c*d^5 - 3*(39*A - 20*B)*a*d
^6)*cos(f*x + e)^3 - (105*B*a*c^5*d - 7*(27*A - 80*B)*a*c^4*d^2 - 6*(150*A - 191*B)*a*c^3*d^3 - 2*(729*A - 610
*B)*a*c^2*d^4 - 3*(340*A - 223*B)*a*c*d^5 - 7*(39*A - 20*B)*a*d^6)*cos(f*x + e)^2 + 2*(35*B*a*c^5*d - 7*(9*A -
20*B)*a*c^4*d^2 - 2*(108*A - 121*B)*a*c^3*d^3 - 2*(141*A - 112*B)*a*c^2*d^4 - (168*A - 107*B)*a*c*d^5 - (39*A
- 20*B)*a*d^6)*cos(f*x + e) + (140*B*a*c^5*d - 28*(9*A - 20*B)*a*c^4*d^2 - 8*(108*A - 121*B)*a*c^3*d^3 - 8*(1
41*A - 112*B)*a*c^2*d^4 - 4*(168*A - 107*B)*a*c*d^5 - 4*(39*A - 20*B)*a*d^6 + (35*B*a*c^3*d^3 - 7*(9*A - 10*B)
*a*c^2*d^4 - (90*A - 67*B)*a*c*d^5 - (39*A - 20*B)*a*d^6)*cos(f*x + e)^4 - 2*(35*B*a*c^4*d^2 - 21*(3*A - 5*B)*
a*c^3*d^3 - (153*A - 137*B)*a*c^2*d^4 - 3*(43*A - 29*B)*a*c*d^5 - (39*A - 20*B)*a*d^6)*cos(f*x + e)^3 - (35*B*
a*c^5*d - 7*(9*A - 40*B)*a*c^4*d^2 - 2*(234*A - 331*B)*a*c^3*d^3 - 2*(447*A - 386*B)*a*c^2*d^4 - (684*A - 455*
B)*a*c*d^5 - 5*(39*A - 20*B)*a*d^6)*cos(f*x + e)^2 + 2*(35*B*a*c^5*d - 7*(9*A - 20*B)*a*c^4*d^2 - 2*(108*A - 1
21*B)*a*c^3*d^3 - 2*(141*A - 112*B)*a*c^2*d^4 - (168*A - 107*B)*a*c*d^5 - (39*A - 20*B)*a*d^6)*cos(f*x + e))*s
in(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2
+ 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*
d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*
d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d
^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x
+ e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*(4*(A - B)*c^6 - 8*(A - B)*c^5*d - 4*(A -
B)*c^4*d^2 + 16*(A - B)*c^3*d^3 - 4*(A - B)*c^2*d^4 - 8*(A - B)*c*d^5 + 4*(A - B)*d^6 - ((3*A + 5*B)*c^4*d^2
- 4*(6*A - 17*B)*c^3*d^3 - 6*(15*A - B)*c^2*d^4 + 4*(12*A - 11*B)*c*d^5 + 7*(9*A - 5*B)*d^6)*cos(f*x + e)^4 -
(2*(3*A + 5*B)*c^5*d - (41*A - 101*B)*c^4*d^2 - 4*(38*A - 31*B)*c^3*d^3 - 2*(39*A + 35*B)*c^2*d^4 + 10*(17*A -
11*B)*c*d^5 + 5*(19*A - 11*B)*d^6)*cos(f*x + e)^3 + ((3*A + 5*B)*c^6 - 16*(A - B)*c^5*d - (31*A - 75*B)*c^4*d
^2 - 4*(21*A - 11*B)*c^3*d^3 - (23*A + 49*B)*c^...

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(co

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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