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

Optimal. Leaf size=309 \[ \frac{\left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{4 f \left (c^2-d^2\right )^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{\left (A d \left (15 c^2+10 c d+7 d^2\right )-B \left (6 c^2 d+3 c^3+19 c d^2+4 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 \sqrt{a} \sqrt{d} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f (c-d)^3} \]

[Out]

-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f))
+ ((A*d*(15*c^2 + 10*c*d + 7*d^2) - B*(3*c^3 + 6*c^2*d + 19*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*Sqrt[a]*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f) - ((B*c - A*d)*Cos
[e + f*x])/(2*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + ((A*d*(7*c + d) - B*(3*c^2 + c*
d + 4*d^2))*Cos[e + f*x])/(4*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

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Rubi [A]  time = 1.05417, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2984, 2985, 2649, 206, 2773, 208} \[ \frac{\left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{4 f \left (c^2-d^2\right )^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{\left (A d \left (15 c^2+10 c d+7 d^2\right )-B \left (6 c^2 d+3 c^3+19 c d^2+4 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 \sqrt{a} \sqrt{d} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f (c-d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f))
+ ((A*d*(15*c^2 + 10*c*d + 7*d^2) - B*(3*c^3 + 6*c^2*d + 19*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*Sqrt[a]*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f) - ((B*c - A*d)*Cos
[e + f*x])/(2*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + ((A*d*(7*c + d) - B*(3*c^2 + c*
d + 4*d^2))*Cos[e + f*x])/(4*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

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

Mathematica [C]  time = 10.6863, size = 847, normalized size = 2.74 \[ \frac{(2+2 i) (A-B) \tanh ^{-1}\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 )}{\left (\sqrt [4]{-1} c^3-3 \sqrt [4]{-1} d c^2+3 \sqrt [4]{-1} d^2 c-\sqrt [4]{-1} d^3\right ) f \sqrt{a (\sin (e+f x)+1)}}-\frac{\left (B \left (3 c^3+6 d c^2+19 d^2 c+4 d^3\right )-A d \left (15 c^2+10 d c+7 d^2\right )\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )-\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{c+d}\right )\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}{16 (c-d)^3 \sqrt{d} (c+d)^{5/2} f \sqrt{a (\sin (e+f x)+1)}}+\frac{\left (B \left (3 c^3+6 d c^2+19 d^2 c+4 d^3\right )-A d \left (15 c^2+10 d c+7 d^2\right )\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (-\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )+\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{c+d}\right )\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}{16 (c-d)^3 \sqrt{d} (c+d)^{5/2} f \sqrt{a (\sin (e+f x)+1)}}+\frac{\left (-3 B \cos \left (\frac{1}{2} (e+f x)\right ) c^2+3 B \sin \left (\frac{1}{2} (e+f x)\right ) c^2+7 A d \cos \left (\frac{1}{2} (e+f x)\right ) c-B d \cos \left (\frac{1}{2} (e+f x)\right ) c-7 A d \sin \left (\frac{1}{2} (e+f x)\right ) c+B d \sin \left (\frac{1}{2} (e+f x)\right ) c+A d^2 \cos \left (\frac{1}{2} (e+f x)\right )-4 B d^2 \cos \left (\frac{1}{2} (e+f x)\right )-A d^2 \sin \left (\frac{1}{2} (e+f x)\right )+4 B d^2 \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}{4 (c-d)^2 (c+d)^2 f \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))}+\frac{\left (-B c \cos \left (\frac{1}{2} (e+f x)\right )+A d \cos \left (\frac{1}{2} (e+f x)\right )+B c \sin \left (\frac{1}{2} (e+f x)\right )-A d \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}{2 (c-d) (c+d) f \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2 + 2*I)*(A - B)*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]))/(((-1)^(1/4)*c^3 - 3*(-1)^(1/4)*c^2*d + 3*(-1)^(1/4)*c*d^2 - (-1)^(1/4)*d^3
)*f*Sqrt[a*(1 + Sin[e + f*x])]) - ((-(A*d*(15*c^2 + 10*c*d + 7*d^2)) + B*(3*c^3 + 6*c^2*d + 19*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] - Sqr
t[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(16*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(
1 + Sin[e + f*x])]) + ((-(A*d*(15*c^2 + 10*c*d + 7*d^2)) + B*(3*c^3 + 6*c^2*d + 19*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]))/(16*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(1 + Sin[e +
f*x])]) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-(B*c*Cos[(e + f*x)/2]) + A*d*Cos[(e + f*x)/2] + B*c*Sin[(e
+ f*x)/2] - A*d*Sin[(e + f*x)/2]))/(2*(c - d)*(c + d)*f*Sqrt[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x])^2) + (
(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-3*B*c^2*Cos[(e + f*x)/2] + 7*A*c*d*Cos[(e + f*x)/2] - B*c*d*Cos[(e + f
*x)/2] + A*d^2*Cos[(e + f*x)/2] - 4*B*d^2*Cos[(e + f*x)/2] + 3*B*c^2*Sin[(e + f*x)/2] - 7*A*c*d*Sin[(e + f*x)/
2] + B*c*d*Sin[(e + f*x)/2] - A*d^2*Sin[(e + f*x)/2] + 4*B*d^2*Sin[(e + f*x)/2]))/(4*(c - d)^2*(c + d)^2*f*Sqr
t[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x]))

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Maple [B]  time = 3.1, size = 2275, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x)

[Out]

1/4*(-16*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^4*c^
2*d^2-4*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^4*c^2*d^2+8*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^4*c^3*d+4*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^4*c^2*d^2-4*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)^2*a^4*d^4+4*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^4*d^4+8*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^4*c*d^3-A*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(
c+d)*d)^(1/2)*c^2*d^2+16*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)*s
in(f*x+e)*a^4*c^2*d^2-4*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)*si
n(f*x+e)^2*a^4*c^2*d^2-8*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)*s
in(f*x+e)^2*a^4*c*d^3+4*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)^2*a^4*c^2*d^2+8*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)*s
in(f*x+e)^2*a^4*c*d^3-8*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)*si
n(f*x+e)*a^4*c^3*d+8*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^4*c^3*d-8*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^4*c*d^3-9*A*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c*d^3+10*A*a^(9/2)*arctanh((-a*(-1+sin(f
*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c*d^4-3*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d
)*d)^(1/2))*sin(f*x+e)^2*c^3*d^2-6*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e
)^2*c^2*d^3-19*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c*d^4+30*A*a^(9/
2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^3*d^2-12*B*a^(9/2)*arctanh((-a*(-1+sin
(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^3*d^2-7*A*a^(5/2)*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/
2)*c^2*d^2+6*A*a^(5/2)*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c*d^3+3*B*a^(5/2)*(-a*(-1+sin(f*x+e)))^(3/
2)*(a*(c+d)*d)^(1/2)*c^3*d-2*B*a^(5/2)*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^2+3*B*a^(5/2)*(-a*(-
1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c*d^3-3*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2
))*c^5-6*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^4*d+20*A*a^(9/2)*arcta
nh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^2*d^3+14*A*a^(9/2)*arctanh((-a*(-1+sin(f*x+e))
)^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c*d^4-4*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^4*c^4+4*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^4*c^4+B*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c^3*d-38*B*a^(9/2)*arctanh((-a*(-1+sin(f*
x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^2*d^3+B*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c^
2*d^2-B*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c*d^3+15*A*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(
1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^2*d^3-8*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1
/2))*sin(f*x+e)*c*d^4+9*A*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c^3*d-8*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^4*c^3*d-6*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e))
)^(1/2)*d/(a*(c+d)*d)^(1/2))*c^4*d-19*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^3*d^
2-4*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^2*d^3+A*a^(7/2)*(-a*(-1+sin(f*x+e)))^(
1/2)*(a*(c+d)*d)^(1/2)*d^4-5*B*a^(7/2)*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*c^4+4*B*a^(7/2)*(-a*(-1+si
n(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*d^4+A*a^(5/2)*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*d^4-4*B*a^(5/2)*
(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*d^4+7*A*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^
(1/2))*sin(f*x+e)^2*d^5-4*B*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*d^5+1
5*A*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^4*d+10*A*a^(9/2)*arctanh((-a*(-1+sin(f*x
+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^3*d^2+7*A*a^(9/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*c
^2*d^3)*(-a*(-1+sin(f*x+e)))^(1/2)*(1+sin(f*x+e))/a^(9/2)/(a*(c+d)*d)^(1/2)/(c+d*sin(f*x+e))^2/(c+d)^2/(c-d)^3
/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))/(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 59.8466, size = 9234, normalized size = 29.88 \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))/(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[1/16*((3*B*c^5 - 3*(5*A - 4*B)*c^4*d - 2*(20*A - 17*B)*c^3*d^2 - 6*(7*A - 8*B)*c^2*d^3 - 3*(8*A - 9*B)*c*d^4
- (7*A - 4*B)*d^5 - (3*B*c^3*d^2 - 3*(5*A - 2*B)*c^2*d^3 - (10*A - 19*B)*c*d^4 - (7*A - 4*B)*d^5)*cos(f*x + e)
^3 - (6*B*c^4*d - 15*(2*A - B)*c^3*d^2 - (35*A - 44*B)*c^2*d^3 - 3*(8*A - 9*B)*c*d^4 - (7*A - 4*B)*d^5)*cos(f*
x + e)^2 + (3*B*c^5 - 3*(5*A - 2*B)*c^4*d - 2*(5*A - 11*B)*c^3*d^2 - 2*(11*A - 5*B)*c^2*d^3 - (10*A - 19*B)*c*
d^4 - (7*A - 4*B)*d^5)*cos(f*x + e) + (3*B*c^5 - 3*(5*A - 4*B)*c^4*d - 2*(20*A - 17*B)*c^3*d^2 - 6*(7*A - 8*B)
*c^2*d^3 - 3*(8*A - 9*B)*c*d^4 - (7*A - 4*B)*d^5 - (3*B*c^3*d^2 - 3*(5*A - 2*B)*c^2*d^3 - (10*A - 19*B)*c*d^4
- (7*A - 4*B)*d^5)*cos(f*x + e)^2 + 2*(3*B*c^4*d - 3*(5*A - 2*B)*c^3*d^2 - (10*A - 19*B)*c^2*d^3 - (7*A - 4*B)
*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*c*d + a*d^2)*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 -
(6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*sqrt(a*c*d + a*d^2)*(d*cos(f*x + e)^2 - (c + 2*d)*cos(f*x + e) + (d*cos
(f*x + e) + c + 3*d)*sin(f*x + e) - c - 3*d)*sqrt(a*sin(f*x + e) + a) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x +
e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*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))) - 8*sqrt(2)*((A - B)*a*c^5*d + 5*(A - B)*a*c^4*d
^2 + 10*(A - B)*a*c^3*d^3 + 10*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 + (A - B)*a*d^6 - ((A - B)*a*c^3*d^3 + 3*
(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^3 - (2*(A - B)*a*c^4*d^2 + 7*(A - B)*a*c^3
*d^3 + 9*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^2 + ((A - B)*a*c^5*d + 3*(A - B)*
a*c^4*d^2 + 4*(A - B)*a*c^3*d^3 + 4*(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e) + ((A
- B)*a*c^5*d + 5*(A - B)*a*c^4*d^2 + 10*(A - B)*a*c^3*d^3 + 10*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 + (A - B)
*a*d^6 - ((A - B)*a*c^3*d^3 + 3*(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^2 + 2*((A
- B)*a*c^4*d^2 + 3*(A - B)*a*c^3*d^3 + 3*(A - B)*a*c^2*d^4 + (A - B)*a*c*d^5)*cos(f*x + e))*sin(f*x + e))*log(
-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*
x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2
))/sqrt(a) + 4*(5*B*c^5*d - (9*A + 2*B)*c^4*d^2 + 2*(3*A - 2*B)*c^3*d^3 + 2*(6*A - B)*c^2*d^4 - (6*A + B)*c*d^
5 - (3*A - 4*B)*d^6 + (3*B*c^4*d^2 - (7*A - B)*c^3*d^3 - (A - B)*c^2*d^4 + (7*A - B)*c*d^5 + (A - 4*B)*d^6)*co
s(f*x + e)^2 + (5*B*c^5*d - (9*A - B)*c^4*d^2 - (A + 3*B)*c^3*d^3 + (11*A - B)*c^2*d^4 + (A - 2*B)*c*d^5 - 2*A
*d^6)*cos(f*x + e) - (5*B*c^5*d - (9*A + 2*B)*c^4*d^2 + 2*(3*A - 2*B)*c^3*d^3 + 2*(6*A - B)*c^2*d^4 - (6*A + B
)*c*d^5 - (3*A - 4*B)*d^6 - (3*B*c^4*d^2 - (7*A - B)*c^3*d^3 - (A - B)*c^2*d^4 + (7*A - B)*c*d^5 + (A - 4*B)*d
^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^6*d^3 - 3*a*c^4*d^5 + 3*a*c^2*d^7 - a*d^9)*f*c
os(f*x + e)^3 + (2*a*c^7*d^2 + a*c^6*d^3 - 6*a*c^5*d^4 - 3*a*c^4*d^5 + 6*a*c^3*d^6 + 3*a*c^2*d^7 - 2*a*c*d^8 -
a*d^9)*f*cos(f*x + e)^2 - (a*c^8*d - 2*a*c^6*d^3 + 2*a*c^2*d^7 - a*d^9)*f*cos(f*x + e) - (a*c^8*d + 2*a*c^7*d
^2 - 2*a*c^6*d^3 - 6*a*c^5*d^4 + 6*a*c^3*d^6 + 2*a*c^2*d^7 - 2*a*c*d^8 - a*d^9)*f + ((a*c^6*d^3 - 3*a*c^4*d^5
+ 3*a*c^2*d^7 - a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^7*d^2 - 3*a*c^5*d^4 + 3*a*c^3*d^6 - a*c*d^8)*f*cos(f*x + e) -
(a*c^8*d + 2*a*c^7*d^2 - 2*a*c^6*d^3 - 6*a*c^5*d^4 + 6*a*c^3*d^6 + 2*a*c^2*d^7 - 2*a*c*d^8 - a*d^9)*f)*sin(f*
x + e)), 1/8*((3*B*c^5 - 3*(5*A - 4*B)*c^4*d - 2*(20*A - 17*B)*c^3*d^2 - 6*(7*A - 8*B)*c^2*d^3 - 3*(8*A - 9*B)
*c*d^4 - (7*A - 4*B)*d^5 - (3*B*c^3*d^2 - 3*(5*A - 2*B)*c^2*d^3 - (10*A - 19*B)*c*d^4 - (7*A - 4*B)*d^5)*cos(f
*x + e)^3 - (6*B*c^4*d - 15*(2*A - B)*c^3*d^2 - (35*A - 44*B)*c^2*d^3 - 3*(8*A - 9*B)*c*d^4 - (7*A - 4*B)*d^5)
*cos(f*x + e)^2 + (3*B*c^5 - 3*(5*A - 2*B)*c^4*d - 2*(5*A - 11*B)*c^3*d^2 - 2*(11*A - 5*B)*c^2*d^3 - (10*A - 1
9*B)*c*d^4 - (7*A - 4*B)*d^5)*cos(f*x + e) + (3*B*c^5 - 3*(5*A - 4*B)*c^4*d - 2*(20*A - 17*B)*c^3*d^2 - 6*(7*A
- 8*B)*c^2*d^3 - 3*(8*A - 9*B)*c*d^4 - (7*A - 4*B)*d^5 - (3*B*c^3*d^2 - 3*(5*A - 2*B)*c^2*d^3 - (10*A - 19*B)
*c*d^4 - (7*A - 4*B)*d^5)*cos(f*x + e)^2 + 2*(3*B*c^4*d - 3*(5*A - 2*B)*c^3*d^2 - (10*A - 19*B)*c^2*d^3 - (7*A
- 4*B)*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*c*d - a*d^2)*arctan(1/2*sqrt(-a*c*d - a*d^2)*sqrt(a*sin(f*x
+ e) + a)*(d*sin(f*x + e) - c - 2*d)/((a*c*d + a*d^2)*cos(f*x + e))) - 4*sqrt(2)*((A - B)*a*c^5*d + 5*(A - B)
*a*c^4*d^2 + 10*(A - B)*a*c^3*d^3 + 10*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 + (A - B)*a*d^6 - ((A - B)*a*c^3*
d^3 + 3*(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^3 - (2*(A - B)*a*c^4*d^2 + 7*(A -
B)*a*c^3*d^3 + 9*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^2 + ((A - B)*a*c^5*d + 3*
(A - B)*a*c^4*d^2 + 4*(A - B)*a*c^3*d^3 + 4*(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e
) + ((A - B)*a*c^5*d + 5*(A - B)*a*c^4*d^2 + 10*(A - B)*a*c^3*d^3 + 10*(A - B)*a*c^2*d^4 + 5*(A - B)*a*c*d^5 +
(A - B)*a*d^6 - ((A - B)*a*c^3*d^3 + 3*(A - B)*a*c^2*d^4 + 3*(A - B)*a*c*d^5 + (A - B)*a*d^6)*cos(f*x + e)^2
+ 2*((A - B)*a*c^4*d^2 + 3*(A - B)*a*c^3*d^3 + 3*(A - B)*a*c^2*d^4 + (A - B)*a*c*d^5)*cos(f*x + e))*sin(f*x +
e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e)
- sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x
+ e) - 2))/sqrt(a) + 2*(5*B*c^5*d - (9*A + 2*B)*c^4*d^2 + 2*(3*A - 2*B)*c^3*d^3 + 2*(6*A - B)*c^2*d^4 - (6*A +
B)*c*d^5 - (3*A - 4*B)*d^6 + (3*B*c^4*d^2 - (7*A - B)*c^3*d^3 - (A - B)*c^2*d^4 + (7*A - B)*c*d^5 + (A - 4*B)
*d^6)*cos(f*x + e)^2 + (5*B*c^5*d - (9*A - B)*c^4*d^2 - (A + 3*B)*c^3*d^3 + (11*A - B)*c^2*d^4 + (A - 2*B)*c*d
^5 - 2*A*d^6)*cos(f*x + e) - (5*B*c^5*d - (9*A + 2*B)*c^4*d^2 + 2*(3*A - 2*B)*c^3*d^3 + 2*(6*A - B)*c^2*d^4 -
(6*A + B)*c*d^5 - (3*A - 4*B)*d^6 - (3*B*c^4*d^2 - (7*A - B)*c^3*d^3 - (A - B)*c^2*d^4 + (7*A - B)*c*d^5 + (A
- 4*B)*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^6*d^3 - 3*a*c^4*d^5 + 3*a*c^2*d^7 - a*
d^9)*f*cos(f*x + e)^3 + (2*a*c^7*d^2 + a*c^6*d^3 - 6*a*c^5*d^4 - 3*a*c^4*d^5 + 6*a*c^3*d^6 + 3*a*c^2*d^7 - 2*a
*c*d^8 - a*d^9)*f*cos(f*x + e)^2 - (a*c^8*d - 2*a*c^6*d^3 + 2*a*c^2*d^7 - a*d^9)*f*cos(f*x + e) - (a*c^8*d + 2
*a*c^7*d^2 - 2*a*c^6*d^3 - 6*a*c^5*d^4 + 6*a*c^3*d^6 + 2*a*c^2*d^7 - 2*a*c*d^8 - a*d^9)*f + ((a*c^6*d^3 - 3*a*
c^4*d^5 + 3*a*c^2*d^7 - a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^7*d^2 - 3*a*c^5*d^4 + 3*a*c^3*d^6 - a*c*d^8)*f*cos(f*
x + e) - (a*c^8*d + 2*a*c^7*d^2 - 2*a*c^6*d^3 - 6*a*c^5*d^4 + 6*a*c^3*d^6 + 2*a*c^2*d^7 - 2*a*c*d^8 - a*d^9)*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))/(c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(1/2),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))/(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError