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

Optimal. Leaf size=402 \[ -\frac{\sqrt{d} \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (10 c^2 d+5 c^3+13 c d^2+4 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 a^{3/2} f (c-d)^4 (c+d)^{5/2}}-\frac{(A (c-13 d)+3 B (c+3 d)) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^4}+\frac{d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{4 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2} \]

[Out]

-((A*(c - 13*d) + 3*B*(c + 3*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2
]*a^(3/2)*(c - d)^4*f) - (Sqrt[d]*(A*d*(35*c^2 + 42*c*d + 19*d^2) - 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))
*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(3/2)*(c - d)^4*(c + d)^
(5/2)*f) - ((A - B)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) + (d*(B*(2*c
+ d) - A*(c + 2*d))*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) +
(d*(3*B*(3*c^2 + 3*c*d + 2*d^2) - A*(2*c^2 + 15*c*d + 7*d^2))*Cos[e + f*x])/(4*a*(c - d)^3*(c + d)^2*f*Sqrt[a
+ a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.56209, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {2978, 2984, 2985, 2649, 206, 2773, 208} \[ -\frac{\sqrt{d} \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (10 c^2 d+5 c^3+13 c d^2+4 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 a^{3/2} f (c-d)^4 (c+d)^{5/2}}-\frac{(A (c-13 d)+3 B (c+3 d)) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^4}+\frac{d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{4 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*(c - 13*d) + 3*B*(c + 3*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2
]*a^(3/2)*(c - d)^4*f) - (Sqrt[d]*(A*d*(35*c^2 + 42*c*d + 19*d^2) - 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))
*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(3/2)*(c - d)^4*(c + d)^
(5/2)*f) - ((A - B)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) + (d*(B*(2*c
+ d) - A*(c + 2*d))*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) +
(d*(3*B*(3*c^2 + 3*c*d + 2*d^2) - A*(2*c^2 + 15*c*d + 7*d^2))*Cos[e + f*x])/(4*a*(c - d)^3*(c + d)^2*f*Sqrt[a
+ a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

Rule 2978

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 2984

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 2985

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]

Rule 2649

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 206

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

Rule 2773

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 208

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

Rubi steps

\begin{align*} \int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx &=-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{\int \frac{-\frac{1}{2} a (A c+3 B c-8 A d+4 B d)-\frac{5}{2} a (A-B) d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx}{2 a^2 (c-d)}\\ &=-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}+\frac{\int \frac{a^2 \left (A \left (c^2-9 c d-7 d^2\right )+3 B \left (c^2+2 c d+2 d^2\right )\right )-3 a^2 d (B (2 c+d)-A (c+2 d)) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{4 a^3 (c-d)^2 (c+d)}\\ &=-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-9 B c^2+15 A c d-9 B c d+7 A d^2-6 B d^2\right ) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac{\int \frac{-\frac{1}{2} a^3 \left (A \left (2 c^3-20 c^2 d-35 c d^2-19 d^3\right )+3 B \left (2 c^3+7 c^2 d+11 c d^2+4 d^3\right )\right )-\frac{1}{2} a^3 d \left (2 A c^2-9 B c^2+15 A c d-9 B c d+7 A d^2-6 B d^2\right ) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{4 a^4 (c-d)^3 (c+d)^2}\\ &=-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-9 B c^2+15 A c d-9 B c d+7 A d^2-6 B d^2\right ) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac{(A (c-13 d)+3 B (c+3 d)) \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx}{4 a (c-d)^4}+\frac{\left (d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 a^2 (c-d)^4 (c+d)^2}\\ &=-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-9 B c^2+15 A c d-9 B c d+7 A d^2-6 B d^2\right ) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac{(A (c-13 d)+3 B (c+3 d)) \operatorname{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 )}{2 a (c-d)^4 f}-\frac{\left (d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right )\right ) \operatorname{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 (c-d)^4 (c+d)^2 f}\\ &=-\frac{(A (c-13 d)+3 B (c+3 d)) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} (c-d)^4 f}-\frac{\sqrt{d} \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 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^{3/2} (c-d)^4 (c+d)^{5/2} f}-\frac{(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-9 B c^2+15 A c d-9 B c d+7 A d^2-6 B d^2\right ) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [C]  time = 13.2433, size = 1395, normalized size = 3.47 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + I)*(A*c + 3*B*c - 13*A*d + 9*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Si
n[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((2*(-1)^(1/4)*c^4 - 8*(-1)^(1/4)*c^3*d + 12*(-1)^(1
/4)*c^2*d^2 - 8*(-1)^(1/4)*c*d^3 + 2*(-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]))^(3/2)) + (Sqrt[d]*(-(A*d*(35*c^2
+ 42*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*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])^3)/(16*(c - d)^4*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(3/2)) - (Sqrt[d]*(-(A*d*(35*c^2 + 4
2*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*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])^3)/(16*(c - d)^4*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(3/2)) + ((Cos[(e + f*x)/2] + Sin[(e + f
*x)/2])*(-8*A*c^4*Cos[(e + f*x)/2] + 8*B*c^4*Cos[(e + f*x)/2] - 8*A*c^3*d*Cos[(e + f*x)/2] + 26*B*c^3*d*Cos[(e
+ f*x)/2] - 22*A*c^2*d^2*Cos[(e + f*x)/2] + 6*B*c^2*d^2*Cos[(e + f*x)/2] - 10*A*c*d^3*Cos[(e + f*x)/2] + 4*B*
c*d^3*Cos[(e + f*x)/2] + 4*B*d^4*Cos[(e + f*x)/2] - 8*A*c^3*d*Cos[(3*(e + f*x))/2] + 26*B*c^3*d*Cos[(3*(e + f*
x))/2] - 40*A*c^2*d^2*Cos[(3*(e + f*x))/2] + 31*B*c^2*d^2*Cos[(3*(e + f*x))/2] - 25*A*c*d^3*Cos[(3*(e + f*x))/
2] + 13*B*c*d^3*Cos[(3*(e + f*x))/2] + A*d^4*Cos[(3*(e + f*x))/2] + 2*B*d^4*Cos[(3*(e + f*x))/2] + 2*A*c^2*d^2
*Cos[(5*(e + f*x))/2] - 9*B*c^2*d^2*Cos[(5*(e + f*x))/2] + 15*A*c*d^3*Cos[(5*(e + f*x))/2] - 9*B*c*d^3*Cos[(5*
(e + f*x))/2] + 7*A*d^4*Cos[(5*(e + f*x))/2] - 6*B*d^4*Cos[(5*(e + f*x))/2] + 8*A*c^4*Sin[(e + f*x)/2] - 8*B*c
^4*Sin[(e + f*x)/2] + 8*A*c^3*d*Sin[(e + f*x)/2] - 26*B*c^3*d*Sin[(e + f*x)/2] + 22*A*c^2*d^2*Sin[(e + f*x)/2]
- 6*B*c^2*d^2*Sin[(e + f*x)/2] + 10*A*c*d^3*Sin[(e + f*x)/2] - 4*B*c*d^3*Sin[(e + f*x)/2] - 4*B*d^4*Sin[(e +
f*x)/2] - 8*A*c^3*d*Sin[(3*(e + f*x))/2] + 26*B*c^3*d*Sin[(3*(e + f*x))/2] - 40*A*c^2*d^2*Sin[(3*(e + f*x))/2]
+ 31*B*c^2*d^2*Sin[(3*(e + f*x))/2] - 25*A*c*d^3*Sin[(3*(e + f*x))/2] + 13*B*c*d^3*Sin[(3*(e + f*x))/2] + A*d
^4*Sin[(3*(e + f*x))/2] + 2*B*d^4*Sin[(3*(e + f*x))/2] - 2*A*c^2*d^2*Sin[(5*(e + f*x))/2] + 9*B*c^2*d^2*Sin[(5
*(e + f*x))/2] - 15*A*c*d^3*Sin[(5*(e + f*x))/2] + 9*B*c*d^3*Sin[(5*(e + f*x))/2] - 7*A*d^4*Sin[(5*(e + f*x))/
2] + 6*B*d^4*Sin[(5*(e + f*x))/2]))/(16*(c - d)^3*(c + d)^2*f*(a*(1 + Sin[e + f*x]))^(3/2)*(c + d*Sin[e + f*x]
)^2)

________________________________________________________________________________________

Maple [B]  time = 3.293, size = 4707, normalized size = 11.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(63*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(
a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^2*d^3-39*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(
a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c*d^4-33*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(
a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^3*d^2-57*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))
*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^2*d^3-3*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2)
)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^3*d^2-15*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/
2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^2*d^3-21*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(
1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c*d^4+60*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(
1/2))*sin(f*x+e)*c^4*d^2+99*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^3*d
^3+90*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^2*d^4+24*B*a^(5/2)*arctan
h((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c*d^5+3*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(
c+d)*d)^(1/2)*sin(f*x+e)*d^5-119*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^
2*c^2*d^4-80*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c*d^5+30*B*a^(5/2)
*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^4*d^2+75*B*a^(5/2)*arctanh((-a*(-1+sin
(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^3*d^3+108*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(
a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^2*d^4-7*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^3-3*B*(-
a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c*d^4-70*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*
(c+d)*d)^(1/2))*sin(f*x+e)^2*c^3*d^3-5*A*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*d^5+4
*B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*d^5+11*A*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)
*(a*(c+d)*d)^(1/2)*c^2*d^3-6*A*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c*d^4-A*2^(1/2)*arctanh(1/
2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^5-11*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2
)*(a*(c+d)*d)^(1/2)*c^3*d^2-A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^3-112*A*a^(5/2)*arcta
nh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^3*d^3-103*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)
))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^2*d^4+2*B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^2*
d^3+B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c*d^4-3*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^
(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^5-4*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(
f*x+e)*d^5+21*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2
*a^2*c^3*d^2+61*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)
^2*a^2*c^2*d^3-6*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e
)^2*a^2*c^4*d-2*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)
^2*a^2*c^4*d-7*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c*d^4-2*A*(-a*(-1+sin(f*x+e))
)^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c^3*d^2+2*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2
)*sin(f*x+e)^2*c^3*d^2+2*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c^2*d^3-2*B*(-a*(
-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c*d^4-51*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)
))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^3*d^2-51*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)
))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^2*d^3-18*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)
))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c*d^4+9*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^
(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^4*d+47*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1
/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^3*d^2+51*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1
/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c*d^4+11*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1
/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^2*d^3+25*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^
(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c*d^4-A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/
2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^3*d^2-21*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(
1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^4*d+26*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/
2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c*d^4-7*B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^
(1/2)*c^3*d^2-4*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*d^5-19*A*a^(5/2)*arctanh((-a*(-1+sin(f*
x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*d^6+12*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*
d)^(1/2))*sin(f*x+e)^2*d^6-42*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^3*d^3-19*A*a
^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^2*d^4+15*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)
))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^5*d-2*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^4*d-35*A*a^(5/2
)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c^2*d^4+13*A*(-a*(-1+sin(f*x+e)))^(1/2)
*a^(3/2)*(a*(c+d)*d)^(1/2)*c*d^4+11*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^4*d+B*(-a*(-1+sin
(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^3*d^2-42*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d
)^(1/2))*sin(f*x+e)^3*c*d^5+15*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*
c^3*d^3+30*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c^2*d^4+39*B*a^(5/2)
*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c*d^5+63*B*a^(5/2)*arctanh((-a*(-1+sin(f
*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c*d^5+2*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)
*sin(f*x+e)^2*d^5-35*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^4*d^2-38*A
*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c*d^5-2*B*(-a*(-1+sin(f*x+e)))^(1/
2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*d^5+15*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1
/2))*sin(f*x+e)*c^5*d-9*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*si
n(f*x+e)^3*a^2*d^5+13*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(
f*x+e)^2*a^2*d^5+11*A*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^3-6*A*(-a*(-1+sin(
f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c*d^4-A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(
1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^5-7*B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*si
n(f*x+e)*c^3*d^2+2*B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^3+B*(-a*(-1+sin(f*x
+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c*d^4-3*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1
/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a^2*c^5+11*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/
a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^4*d+25*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(
c+d)*d)^(1/2)*a^2*c^3*d^2-9*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2
)*sin(f*x+e)^2*a^2*d^5-15*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*
a^2*c^4*d-21*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^3*d^2-9
*B*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^2*d^3+2*A*(-a*(-1+s
in(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c*d^4-2*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)
*d)^(1/2)*sin(f*x+e)^2*c^2*d^3-17*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^3*d^2+A*
(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^3+13*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f
*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*d^5-4*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(
a*(c+d)*d)^(1/2)*sin(f*x+e)*c^4*d+13*A*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d
)*d)^(1/2)*a^2*c^2*d^3+17*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c*d^4+13*B*(-a*(-1
+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^4*d+7*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)
*d)^(1/2)*sin(f*x+e)*c^3*d^2-9*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^3+30*B*
a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^4*d^2+39*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e
)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^3*d^3+12*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^
2*d^4-2*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^5+3*A*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(
c+d)*d)^(1/2)*d^5-35*A*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^4*d^2-19*A*a^(5/2)*ar
ctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*d^6+12*B*a^(5/2)*arctanh((-a*(-1+sin(f*x+e)
))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*d^6-5*A*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*d^5+4*
B*(-a*(-1+sin(f*x+e)))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*d^5+2*B*(-a*(-1+sin(f*x+e)))^(1/2)*a^(3/2)*(a*(c+d)*d)^
(1/2)*c^5)/(a*(c+d)*d)^(1/2)/(c+d*sin(f*x+e))^2/(c+d)^2/(c-d)^4/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [B]  time = 46.4561, size = 13168, normalized size = 32.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(2*sqrt(2)*(2*(A + 3*B)*c^5 - 6*(3*A - 7*B)*c^4*d - 4*(23*A - 27*B)*c^3*d^2 - 4*(37*A - 33*B)*c^2*d^3 -
6*(17*A - 13*B)*c*d^4 - 2*(13*A - 9*B)*d^5 + ((A + 3*B)*c^3*d^2 - (11*A - 15*B)*c^2*d^3 - (25*A - 21*B)*c*d^4
- (13*A - 9*B)*d^5)*cos(f*x + e)^4 - (2*(A + 3*B)*c^4*d - 3*(7*A - 11*B)*c^3*d^2 - (61*A - 57*B)*c^2*d^3 - 3*(
17*A - 13*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^3 - ((A + 3*B)*c^5 - (7*A - 27*B)*c^4*d - 6*(11*A - 15*B)*
c^3*d^2 - 2*(73*A - 69*B)*c^2*d^3 - (127*A - 99*B)*c*d^4 - 3*(13*A - 9*B)*d^5)*cos(f*x + e)^2 + ((A + 3*B)*c^5
- 3*(3*A - 7*B)*c^4*d - 2*(23*A - 27*B)*c^3*d^2 - 2*(37*A - 33*B)*c^2*d^3 - 3*(17*A - 13*B)*c*d^4 - (13*A - 9
*B)*d^5)*cos(f*x + e) + (2*(A + 3*B)*c^5 - 6*(3*A - 7*B)*c^4*d - 4*(23*A - 27*B)*c^3*d^2 - 4*(37*A - 33*B)*c^2
*d^3 - 6*(17*A - 13*B)*c*d^4 - 2*(13*A - 9*B)*d^5 - ((A + 3*B)*c^3*d^2 - (11*A - 15*B)*c^2*d^3 - (25*A - 21*B)
*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^3 - 2*((A + 3*B)*c^4*d - 2*(5*A - 9*B)*c^3*d^2 - 36*(A - B)*c^2*d^3 -
2*(19*A - 15*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^2 + ((A + 3*B)*c^5 - 3*(3*A - 7*B)*c^4*d - 2*(23*A - 27
*B)*c^3*d^2 - 2*(37*A - 33*B)*c^2*d^3 - 3*(17*A - 13*B)*c*d^4 - (13*A - 9*B)*d^5)*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)) + (30*B*a*c^5 - 10*(7*A - 12*B)*a*c^4*d - 4*(56*A - 57*B)*a*c^3*d^2 - 12*(23*A - 20*B
)*a*c^2*d^3 - 2*(80*A - 63*B)*a*c*d^4 - 2*(19*A - 12*B)*a*d^5 + (15*B*a*c^3*d^2 - 5*(7*A - 6*B)*a*c^2*d^3 - 3*
(14*A - 13*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)^4 - (30*B*a*c^4*d - 5*(14*A - 15*B)*a*c^3*d^2 - (119
*A - 108*B)*a*c^2*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)^3 - (15*B*a*c^5 - 5*(7*A - 1
8*B)*a*c^4*d - 2*(91*A - 102*B)*a*c^3*d^2 - 2*(146*A - 129*B)*a*c^2*d^3 - (202*A - 165*B)*a*c*d^4 - 3*(19*A -
12*B)*a*d^5)*cos(f*x + e)^2 + (15*B*a*c^5 - 5*(7*A - 12*B)*a*c^4*d - 2*(56*A - 57*B)*a*c^3*d^2 - 6*(23*A - 20*
B)*a*c^2*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e) + (30*B*a*c^5 - 10*(7*A - 12*B)*a*c^4
*d - 4*(56*A - 57*B)*a*c^3*d^2 - 12*(23*A - 20*B)*a*c^2*d^3 - 2*(80*A - 63*B)*a*c*d^4 - 2*(19*A - 12*B)*a*d^5
- (15*B*a*c^3*d^2 - 5*(7*A - 6*B)*a*c^2*d^3 - 3*(14*A - 13*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)^3 -
2*(15*B*a*c^4*d - 5*(7*A - 9*B)*a*c^3*d^2 - (77*A - 69*B)*a*c^2*d^3 - (61*A - 51*B)*a*c*d^4 - (19*A - 12*B)*a*
d^5)*cos(f*x + e)^2 + (15*B*a*c^5 - 5*(7*A - 12*B)*a*c^4*d - 2*(56*A - 57*B)*a*c^3*d^2 - 6*(23*A - 20*B)*a*c^2
*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*c
os(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*(2*(A - B)*c^5 - 2*(A - B)*c^4*d - 4*(A - B)*c^3*d^2 + 4*(A - B)*c^2*d^3 + 2*(A - B)
*c*d^4 - 2*(A - B)*d^5 - ((2*A - 9*B)*c^3*d^2 + 13*A*c^2*d^3 - (8*A - 3*B)*c*d^4 - (7*A - 6*B)*d^5)*cos(f*x +
e)^3 + ((4*A - 13*B)*c^4*d + (15*A + 2*B)*c^3*d^2 - (14*A - 9*B)*c^2*d^3 - (9*A - 4*B)*c*d^4 + 2*(2*A - B)*d^5
)*cos(f*x + e)^2 + (2*(A - B)*c^5 + (2*A - 11*B)*c^4*d + (13*A - 3*B)*c^3*d^2 + (3*A + 5*B)*c^2*d^3 - 5*(3*A -
B)*c*d^4 - (5*A - 6*B)*d^5)*cos(f*x + e) - (2*(A - B)*c^5 - 2*(A - B)*c^4*d - 4*(A - B)*c^3*d^2 + 4*(A - B)*c
^2*d^3 + 2*(A - B)*c*d^4 - 2*(A - B)*d^5 - ((2*A - 9*B)*c^3*d^2 + 13*A*c^2*d^3 - (8*A - 3*B)*c*d^4 - (7*A - 6*
B)*d^5)*cos(f*x + e)^2 - ((4*A - 13*B)*c^4*d + (17*A - 7*B)*c^3*d^2 - (A - 9*B)*c^2*d^3 - (17*A - 7*B)*c*d^4 -
(3*A - 4*B)*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^
4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3*a^2*c^6*d^2 -
4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^2*c^8 + 2*a^2*
c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*c*d^7 + 3*a^2*
d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) + 2
*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c
^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d - a^2*c^6*d^2 -
3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^2 - (a^2*c
^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*
a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e)), 1/8*(sqrt(2)*(2*(A + 3*B)*c^5 - 6*(3*A - 7*B)*c^4*d -
4*(23*A - 27*B)*c^3*d^2 - 4*(37*A - 33*B)*c^2*d^3 - 6*(17*A - 13*B)*c*d^4 - 2*(13*A - 9*B)*d^5 + ((A + 3*B)*c
^3*d^2 - (11*A - 15*B)*c^2*d^3 - (25*A - 21*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^4 - (2*(A + 3*B)*c^4*d -
3*(7*A - 11*B)*c^3*d^2 - (61*A - 57*B)*c^2*d^3 - 3*(17*A - 13*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^3 - (
(A + 3*B)*c^5 - (7*A - 27*B)*c^4*d - 6*(11*A - 15*B)*c^3*d^2 - 2*(73*A - 69*B)*c^2*d^3 - (127*A - 99*B)*c*d^4
- 3*(13*A - 9*B)*d^5)*cos(f*x + e)^2 + ((A + 3*B)*c^5 - 3*(3*A - 7*B)*c^4*d - 2*(23*A - 27*B)*c^3*d^2 - 2*(37*
A - 33*B)*c^2*d^3 - 3*(17*A - 13*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e) + (2*(A + 3*B)*c^5 - 6*(3*A - 7*B)*
c^4*d - 4*(23*A - 27*B)*c^3*d^2 - 4*(37*A - 33*B)*c^2*d^3 - 6*(17*A - 13*B)*c*d^4 - 2*(13*A - 9*B)*d^5 - ((A +
3*B)*c^3*d^2 - (11*A - 15*B)*c^2*d^3 - (25*A - 21*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^3 - 2*((A + 3*B)*
c^4*d - 2*(5*A - 9*B)*c^3*d^2 - 36*(A - B)*c^2*d^3 - 2*(19*A - 15*B)*c*d^4 - (13*A - 9*B)*d^5)*cos(f*x + e)^2
+ ((A + 3*B)*c^5 - 3*(3*A - 7*B)*c^4*d - 2*(23*A - 27*B)*c^3*d^2 - 2*(37*A - 33*B)*c^2*d^3 - 3*(17*A - 13*B)*c
*d^4 - (13*A - 9*B)*d^5)*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)) + (30*B*a*c^5 - 10*(7*A - 12*B)*
a*c^4*d - 4*(56*A - 57*B)*a*c^3*d^2 - 12*(23*A - 20*B)*a*c^2*d^3 - 2*(80*A - 63*B)*a*c*d^4 - 2*(19*A - 12*B)*a
*d^5 + (15*B*a*c^3*d^2 - 5*(7*A - 6*B)*a*c^2*d^3 - 3*(14*A - 13*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)
^4 - (30*B*a*c^4*d - 5*(14*A - 15*B)*a*c^3*d^2 - (119*A - 108*B)*a*c^2*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 1
2*B)*a*d^5)*cos(f*x + e)^3 - (15*B*a*c^5 - 5*(7*A - 18*B)*a*c^4*d - 2*(91*A - 102*B)*a*c^3*d^2 - 2*(146*A - 12
9*B)*a*c^2*d^3 - (202*A - 165*B)*a*c*d^4 - 3*(19*A - 12*B)*a*d^5)*cos(f*x + e)^2 + (15*B*a*c^5 - 5*(7*A - 12*B
)*a*c^4*d - 2*(56*A - 57*B)*a*c^3*d^2 - 6*(23*A - 20*B)*a*c^2*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 12*B)*a*d^
5)*cos(f*x + e) + (30*B*a*c^5 - 10*(7*A - 12*B)*a*c^4*d - 4*(56*A - 57*B)*a*c^3*d^2 - 12*(23*A - 20*B)*a*c^2*d
^3 - 2*(80*A - 63*B)*a*c*d^4 - 2*(19*A - 12*B)*a*d^5 - (15*B*a*c^3*d^2 - 5*(7*A - 6*B)*a*c^2*d^3 - 3*(14*A - 1
3*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)^3 - 2*(15*B*a*c^4*d - 5*(7*A - 9*B)*a*c^3*d^2 - (77*A - 69*B)
*a*c^2*d^3 - (61*A - 51*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f*x + e)^2 + (15*B*a*c^5 - 5*(7*A - 12*B)*a*c^4*
d - 2*(56*A - 57*B)*a*c^3*d^2 - 6*(23*A - 20*B)*a*c^2*d^3 - (80*A - 63*B)*a*c*d^4 - (19*A - 12*B)*a*d^5)*cos(f
*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqr
t(-d/(a*c + a*d))/(d*cos(f*x + e))) - 2*(2*(A - B)*c^5 - 2*(A - B)*c^4*d - 4*(A - B)*c^3*d^2 + 4*(A - B)*c^2*d
^3 + 2*(A - B)*c*d^4 - 2*(A - B)*d^5 - ((2*A - 9*B)*c^3*d^2 + 13*A*c^2*d^3 - (8*A - 3*B)*c*d^4 - (7*A - 6*B)*d
^5)*cos(f*x + e)^3 + ((4*A - 13*B)*c^4*d + (15*A + 2*B)*c^3*d^2 - (14*A - 9*B)*c^2*d^3 - (9*A - 4*B)*c*d^4 + 2
*(2*A - B)*d^5)*cos(f*x + e)^2 + (2*(A - B)*c^5 + (2*A - 11*B)*c^4*d + (13*A - 3*B)*c^3*d^2 + (3*A + 5*B)*c^2*
d^3 - 5*(3*A - B)*c*d^4 - (5*A - 6*B)*d^5)*cos(f*x + e) - (2*(A - B)*c^5 - 2*(A - B)*c^4*d - 4*(A - B)*c^3*d^2
+ 4*(A - B)*c^2*d^3 + 2*(A - B)*c*d^4 - 2*(A - B)*d^5 - ((2*A - 9*B)*c^3*d^2 + 13*A*c^2*d^3 - (8*A - 3*B)*c*d
^4 - (7*A - 6*B)*d^5)*cos(f*x + e)^2 - ((4*A - 13*B)*c^4*d + (17*A - 7*B)*c^3*d^2 - (A - 9*B)*c^2*d^3 - (17*A
- 7*B)*c*d^4 - (3*A - 4*B)*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^6*d^2 - 2*a^2*c^
5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3
*a^2*c^6*d^2 - 4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^
2*c^8 + 2*a^2*c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*
c*d^7 + 3*a^2*d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*co
s(f*x + e) + 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c
^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d -
a^2*c^6*d^2 - 3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x +
e)^2 - (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^
2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError