\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx\) [326]

Optimal result

Integrand size = 37, antiderivative size = 395 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=-\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^4 f}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} (c-d)^4 (c+d)^{3/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac {(3 A c+5 B c-15 A d+7 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]

[Out]

d^(3/2)*(A*d*(7*c+5*d)-B*(5*c^2+5*c*d+2*d^2))*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)^4/(c+d)^(3/2)/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))-1/16
*(3*A*c-15*A*d+5*B*c+7*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))-1/32*(B*(5*c^2-58*c
*d-43*d^2)+A*(3*c^2-22*c*d+115*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)^4/f*2^(1/2)-1/16*d*(A*(3*c^2-16*c*d-35*d^2)+B*(5*c^2+32*c*d+11*d^2))*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*s
in(f*x+e))/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3057, 3063, 3064, 2728, 212, 2852, 214} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=-\frac {\left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^4}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {(3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))} \]

[In]

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

[Out]

-1/16*((B*(5*c^2 - 58*c*d - 43*d^2) + A*(3*c^2 - 22*c*d + 115*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)^4*f) + (d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2
))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^
(3/2)*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])) - ((3*A*c + 5*
B*c - 15*A*d + 7*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])) - (d*(A
*(3*c^2 - 16*c*d - 35*d^2) + B*(5*c^2 + 32*c*d + 11*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]))

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))}-\frac {\int \frac {-\frac {1}{2} a (3 A c+5 B c-10 A d+2 B d)-\frac {5}{2} a (A-B) d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2} \, 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))}-\frac {(3 A c+5 B c-15 A d+7 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))}+\frac {\int \frac {\frac {1}{4} a^2 \left (B \left (5 c^2-43 c d-22 d^2\right )+A \left (3 c^2-13 c d+70 d^2\right )\right )+\frac {3}{4} a^2 d (3 A c+5 B c-15 A d+7 B d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, 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))}-\frac {(3 A c+5 B c-15 A d+7 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))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 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))}-\frac {\int \frac {-\frac {1}{4} a^3 \left (B \left (5 c^3-48 c^2 d-69 c d^2-32 d^3\right )+A \left (3 c^3-16 c^2 d+77 c d^2+80 d^3\right )\right )-\frac {1}{4} a^3 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{8 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))}-\frac {(3 A c+5 B c-15 A d+7 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))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 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))}-\frac {\left (d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 a^3 (c-d)^4 (c+d)}+\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 d^2\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2 (c-d)^4} \\ & = -\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))}-\frac {(3 A c+5 B c-15 A d+7 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))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 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))}+\frac {\left (d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\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 )}{a^2 (c-d)^4 (c+d) f}-\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 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)^4 f} \\ & = -\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 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)^4 f}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\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 )}{a^{5/2} (c-d)^4 (c+d)^{3/2} 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))}-\frac {(3 A c+5 B c-15 A d+7 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))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 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))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 15.73 (sec) , antiderivative size = 1680, normalized size of antiderivative = 4.25 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\frac {(1+i) \left (3 A c^2+5 B c^2-22 A c d-58 B c d+115 A d^2-43 B d^2\right ) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {1}{4} (e+f x)\right ) \left (\cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{\left (16 \sqrt [4]{-1} c^4-64 \sqrt [4]{-1} c^3 d+96 \sqrt [4]{-1} c^2 d^2-64 \sqrt [4]{-1} c d^3+16 \sqrt [4]{-1} d^4\right ) f (a (1+\sin (e+f x)))^{5/2}}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+\text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )+\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+2 \sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+3 d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{4 (c-d)^4 (c+d)^{3/2} f (a (1+\sin (e+f x)))^{5/2}}+\frac {d^{3/2} \left (-A d (7 c+5 d)+B \left (5 c^2+5 c d+2 d^2\right )\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+\text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}-2 \sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+3 d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{4 (c-d)^4 (c+d)^{3/2} f (a (1+\sin (e+f x)))^{5/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-22 A c^3 \cos \left (\frac {1}{2} (e+f x)\right )+6 B c^3 \cos \left (\frac {1}{2} (e+f x)\right )+40 A c^2 d \cos \left (\frac {1}{2} (e+f x)\right )-40 B c^2 d \cos \left (\frac {1}{2} (e+f x)\right )+54 A c d^2 \cos \left (\frac {1}{2} (e+f x)\right )-70 B c d^2 \cos \left (\frac {1}{2} (e+f x)\right )+24 A d^3 \cos \left (\frac {1}{2} (e+f x)\right )+8 B d^3 \cos \left (\frac {1}{2} (e+f x)\right )-6 A c^3 \cos \left (\frac {3}{2} (e+f x)\right )-10 B c^3 \cos \left (\frac {3}{2} (e+f x)\right )+21 A c^2 d \cos \left (\frac {3}{2} (e+f x)\right )-29 B c^2 d \cos \left (\frac {3}{2} (e+f x)\right )+54 A c d^2 \cos \left (\frac {3}{2} (e+f x)\right )-86 B c d^2 \cos \left (\frac {3}{2} (e+f x)\right )+75 A d^3 \cos \left (\frac {3}{2} (e+f x)\right )-19 B d^3 \cos \left (\frac {3}{2} (e+f x)\right )+3 A c^2 d \cos \left (\frac {5}{2} (e+f x)\right )+5 B c^2 d \cos \left (\frac {5}{2} (e+f x)\right )-16 A c d^2 \cos \left (\frac {5}{2} (e+f x)\right )+32 B c d^2 \cos \left (\frac {5}{2} (e+f x)\right )-35 A d^3 \cos \left (\frac {5}{2} (e+f x)\right )+11 B d^3 \cos \left (\frac {5}{2} (e+f x)\right )+22 A c^3 \sin \left (\frac {1}{2} (e+f x)\right )-6 B c^3 \sin \left (\frac {1}{2} (e+f x)\right )-40 A c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+40 B c^2 d \sin \left (\frac {1}{2} (e+f x)\right )-54 A c d^2 \sin \left (\frac {1}{2} (e+f x)\right )+70 B c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-24 A d^3 \sin \left (\frac {1}{2} (e+f x)\right )-8 B d^3 \sin \left (\frac {1}{2} (e+f x)\right )-6 A c^3 \sin \left (\frac {3}{2} (e+f x)\right )-10 B c^3 \sin \left (\frac {3}{2} (e+f x)\right )+21 A c^2 d \sin \left (\frac {3}{2} (e+f x)\right )-29 B c^2 d \sin \left (\frac {3}{2} (e+f x)\right )+54 A c d^2 \sin \left (\frac {3}{2} (e+f x)\right )-86 B c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+75 A d^3 \sin \left (\frac {3}{2} (e+f x)\right )-19 B d^3 \sin \left (\frac {3}{2} (e+f x)\right )-3 A c^2 d \sin \left (\frac {5}{2} (e+f x)\right )-5 B c^2 d \sin \left (\frac {5}{2} (e+f x)\right )+16 A c d^2 \sin \left (\frac {5}{2} (e+f x)\right )-32 B c d^2 \sin \left (\frac {5}{2} (e+f x)\right )+35 A d^3 \sin \left (\frac {5}{2} (e+f x)\right )-11 B d^3 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{64 (c-d)^3 (c+d) f (a (1+\sin (e+f x)))^{5/2} (c+d \sin (e+f x))} \]

[In]

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

[Out]

((1 + I)*(3*A*c^2 + 5*B*c^2 - 22*A*c*d - 58*B*c*d + 115*A*d^2 - 43*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^
4 - 64*(-1)^(1/4)*c^3*d + 96*(-1)^(1/4)*c^2*d^2 - 64*(-1)^(1/4)*c*d^3 + 16*(-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f
*x]))^(5/2)) + (d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + R
ootSum[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]*L
og[-#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)/(4*
(c - d)^4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-(A*d*(7*c + 5*d)) + B*(5*c^2 + 5*c*d + 2*
d^2))*(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)/(4*(c - d)^4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[
(e + f*x)/2] + Sin[(e + f*x)/2])*(-22*A*c^3*Cos[(e + f*x)/2] + 6*B*c^3*Cos[(e + f*x)/2] + 40*A*c^2*d*Cos[(e +
f*x)/2] - 40*B*c^2*d*Cos[(e + f*x)/2] + 54*A*c*d^2*Cos[(e + f*x)/2] - 70*B*c*d^2*Cos[(e + f*x)/2] + 24*A*d^3*C
os[(e + f*x)/2] + 8*B*d^3*Cos[(e + f*x)/2] - 6*A*c^3*Cos[(3*(e + f*x))/2] - 10*B*c^3*Cos[(3*(e + f*x))/2] + 21
*A*c^2*d*Cos[(3*(e + f*x))/2] - 29*B*c^2*d*Cos[(3*(e + f*x))/2] + 54*A*c*d^2*Cos[(3*(e + f*x))/2] - 86*B*c*d^2
*Cos[(3*(e + f*x))/2] + 75*A*d^3*Cos[(3*(e + f*x))/2] - 19*B*d^3*Cos[(3*(e + f*x))/2] + 3*A*c^2*d*Cos[(5*(e +
f*x))/2] + 5*B*c^2*d*Cos[(5*(e + f*x))/2] - 16*A*c*d^2*Cos[(5*(e + f*x))/2] + 32*B*c*d^2*Cos[(5*(e + f*x))/2]
- 35*A*d^3*Cos[(5*(e + f*x))/2] + 11*B*d^3*Cos[(5*(e + f*x))/2] + 22*A*c^3*Sin[(e + f*x)/2] - 6*B*c^3*Sin[(e +
f*x)/2] - 40*A*c^2*d*Sin[(e + f*x)/2] + 40*B*c^2*d*Sin[(e + f*x)/2] - 54*A*c*d^2*Sin[(e + f*x)/2] + 70*B*c*d^
2*Sin[(e + f*x)/2] - 24*A*d^3*Sin[(e + f*x)/2] - 8*B*d^3*Sin[(e + f*x)/2] - 6*A*c^3*Sin[(3*(e + f*x))/2] - 10*
B*c^3*Sin[(3*(e + f*x))/2] + 21*A*c^2*d*Sin[(3*(e + f*x))/2] - 29*B*c^2*d*Sin[(3*(e + f*x))/2] + 54*A*c*d^2*Si
n[(3*(e + f*x))/2] - 86*B*c*d^2*Sin[(3*(e + f*x))/2] + 75*A*d^3*Sin[(3*(e + f*x))/2] - 19*B*d^3*Sin[(3*(e + f*
x))/2] - 3*A*c^2*d*Sin[(5*(e + f*x))/2] - 5*B*c^2*d*Sin[(5*(e + f*x))/2] + 16*A*c*d^2*Sin[(5*(e + f*x))/2] - 3
2*B*c*d^2*Sin[(5*(e + f*x))/2] + 35*A*d^3*Sin[(5*(e + f*x))/2] - 11*B*d^3*Sin[(5*(e + f*x))/2]))/(64*(c - d)^3
*(c + d)*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4091\) vs. \(2(358)=716\).

Time = 2.02 (sec) , antiderivative size = 4092, normalized size of antiderivative = 10.36

method	result	size
default	\(\text {Expression too large to display}\)	\(4092\)

[In]

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

[Out]

-1/32*(-84*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^2*d^2+6*A*(a*(c+d)*d)^(1/2)*2^(1
/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^4+115*A*(a*(c+d)*d)^(1/2)*2^(1/2)*
arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*d^4-43*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arcta
nh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*d^4+3*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1
/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^4-43*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*
(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*d^4-19*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(
sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3*d+93*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))
^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d^2+115*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1
/2)/a^(1/2))*a^2*c*d^3-53*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a
^2*c^3*d-101*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d^2-43
*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c*d^3-10*B*(-a*(sin(f*
x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^3*d-22*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(
1/2)*sin(f*x+e)*c^2*d^2-76*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c*d^3-32*B*(-a*(si
n(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c*d^3+20*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/
2)*a^(3/2)*sin(f*x+e)*c^3*d-84*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c*d^3-32*A*(-a
*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c*d^3+32*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)
^(1/2)*a^(3/2)*sin(f*x+e)^2*c^2*d^2+10*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/
2)/a^(1/2))*sin(f*x+e)*a^2*c^4+115*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a
^(1/2))*sin(f*x+e)^3*a^2*d^4+10*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c*d^3-86*B*(a
*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*d^4-6*A*(-a*(s
in(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^3*d+38*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2
)*a^(1/2)*sin(f*x+e)*c^2*d^2+6*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c*d^3+230*A*(a
*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*d^4+5*B*(a*(c+
d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^4+12*B*(-a*(sin(
f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^3*d+116*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*
a^(3/2)*sin(f*x+e)*c^2*d^2-160*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*d
^5+64*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*d^5-160*A*arctanh((-a*(sin
(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*d^5+32*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)
*a^(3/2)*d^4+12*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^4-320*A*arctanh((-a*(sin(f*x+e)-1))^(1
/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*d^5+128*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))
*a^(5/2)*sin(f*x+e)^2*d^5+64*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*d^5+1
60*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c^3*d^2+160*B*arctanh((-a*(sin(f*x+e)-1))^
(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c^2*d^3-6*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^4-10*B*(-
a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^4-160*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1
/2))*a^(5/2)*c*d^4+64*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c*d^4+20*A*(-a*(sin(f*x
+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^4-224*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/
2)*c^2*d^3-101*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a
^2*c^3*d-255*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2
*c^2*d^2-187*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2
*c*d^3-207*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2
*c^2*d^2-245*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a
^2*c*d^3-35*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*
c^3*d+93*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c
*d^3+5*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^3
*d-53*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^2*
d^2-101*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c*
d^3-13*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^3
*d+55*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^2*
d^2+3*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^3*
d-19*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^2*d
^2+167*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^2*d
^2+323*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c*d^3
+301*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c*d^3
-43*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^3*d+
160*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^3*d^2+480*B*arctanh((-a*(s
in(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^2*d^3+384*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*
d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c*d^4+22*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c*d^3
+22*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*d^4+38*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+
d)*d)^(1/2)*a^(1/2)*c^3*d+5*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))
*a^2*c^4-84*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^3*d-20*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d
)*d)^(1/2)*a^(3/2)*c^2*d^2-38*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*d^4+6*A*(-a*(si
n(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^2*d^2-38*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*
c*d^3+32*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*d^4+160*B*arctanh((-a*(sin(f*x+e)-
1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c^2*d^3+160*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)
*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c*d^4+52*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^3*d+20*B*(-a*
(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^2*d^2-84*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/
2)*c*d^3+320*B*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^3*d^2+480*B*arctanh
((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^2*d^3+288*B*arctanh((-a*(sin(f*x+e)-1))^(
1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c*d^4+148*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*s
in(f*x+e)*d^4-52*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*d^4-448*A*arctanh((-a*(sin(f
*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^2*d^3-544*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(
c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c*d^4-224*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*s
in(f*x+e)^3*c*d^4+3*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^4
-22*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^3*d+10*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/
2)*a^(1/2)*c^2*d^2-224*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^2*d^3-6
08*A*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c*d^4+52*A*(-a*(sin(f*x+e)-1)
)^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c*d^3)*(-a*(sin(f*x+e)-1))^(1/2)/a^(9/2)/(1+sin(f*x+e))/(a*(c+d)*d)^(1/2)/(c
+d*sin(f*x+e))/(c+d)/(c-d)^4/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2433 vs. \(2 (358) = 716\).

Time = 8.10 (sec) , antiderivative size = 5151, normalized size of antiderivative = 13.04 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (358) = 716\).

Time = 0.71 (sec) , antiderivative size = 1099, normalized size of antiderivative = 2.78 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(16*sqrt(2)*(5*sqrt(2)*B*sqrt(a)*c^2*d^2 - 7*sqrt(2)*A*sqrt(a)*c*d^3 + 5*sqrt(2)*B*sqrt(a)*c*d^3 - 5*sqr
t(2)*A*sqrt(a)*d^4 + 2*sqrt(2)*B*sqrt(a)*d^4)*arctan(sqrt(2)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)/sqrt(-c*d - d^2)
)/((a^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*a^3*c^3*
d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*a^3*c*d^4*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a^3*d^5*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 - 22*A*sqrt(a)*c*d - 58*B*sqrt(a)*c*d + 115*A*sqrt(a)*d^2 - 43*B*sqrt(a)*d^2)*log(sin(
-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c^3*d*sgn
(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2)*a^3*c^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c
*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + (3*A*sqrt(a)
*c^2 + 5*B*sqrt(a)*c^2 - 22*A*sqrt(a)*c*d - 58*B*sqrt(a)*c*d + 115*A*sqrt(a)*d^2 - 43*B*sqrt(a)*d^2)*log(-sin(
-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c^3*d*sgn
(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2)*a^3*c^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c
*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + 64*(B*sqrt(a
)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - A*sqrt(a)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/((sqrt(2)*a^3*c^4*sgn(c
os(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*sqrt(2)*a^3*c^3*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*sqrt(2)*a^3*c*d^3
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*d^4*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)^2 - c - d)) + 2*(3*A*sqrt(a)*c*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 5*B*sqrt(a)*c*sin(-1/4*pi
+ 1/2*f*x + 1/2*e)^3 - 19*A*sqrt(a)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 11*B*sqrt(a)*d*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)^3 - 5*A*sqrt(a)*c*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3*B*sqrt(a)*c*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 21*
A*sqrt(a)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 13*B*sqrt(a)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e))/((sqrt(2)*a^3*c^3*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*sqrt(2)*a^3*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*sqrt(2)*a^3*
c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*(sin(-1/4*pi
+ 1/2*f*x + 1/2*e)^2 - 1)^2))/f

Mupad [F(-1)]

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

[In]

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

[Out]

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