∫ 1____________∘ a+a sin(e+fx) (c+d sin(e+fx))3 dx

3.549 \(\int \frac {1}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=247 \[ \frac {d (7 c+d) \cos (e+f x)}{4 f \left (c^2-d^2\right )^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {d \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac {\sqrt {d} \left (15 c^2+10 c d+7 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 \sqrt {a} f (c-d)^3 (c+d)^{5/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f (c-d)^3} \]

[Out]

-arctanh(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))*2^(1/2)/(c-d)^3/f/a^(1/2)+1/4*(15*c^2+10*c*d+7

*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))*d^(1/2)/(c-d)^3/(c+d)^(5/2)/f/a^(

1/2)+1/2*d*cos(f*x+e)/(c^2-d^2)/f/(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2)+1/4*d*(7*c+d)*cos(f*x+e)/(c^2-d^2)

^2/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A] time = 0.73, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2779, 2984, 2985, 2649, 206, 2773, 208} \[ \frac {d (7 c+d) \cos (e+f x)}{4 f \left (c^2-d^2\right )^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {d \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac {\sqrt {d} \left (15 c^2+10 c d+7 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 \sqrt {a} f (c-d)^3 (c+d)^{5/2}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f (c-d)^3} \]

Antiderivative was successfully veriﬁed.

[In]

Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3),x]

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f)) + (Sqrt[

d]*(15*c^2 + 10*c*d + 7*d^2)*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*(c + d)^(5/2)*f) + (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) + (d*(7*c + d)*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 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 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]

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 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 2779

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> -Sim

p[(d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/

(2*b*(n + 1)*(c^2 - d^2)), Int[((c + d*Sin[e + f*x])^(n + 1)*Simp[a*d - 2*b*c*(n + 1) + b*d*(2*n + 3)*Sin[e +

f*x], x])/Sqrt[a + b*Sin[e + f*x]], 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] && LtQ[n, -1] && IntegerQ[2*n]

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx &=\frac {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 {a (4 c+d)-3 a d \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{4 a \left (c^2-d^2\right )}\\ &=\frac {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 {d (7 c+d) \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}{2} a^2 \left (8 c^2+9 c d+7 d^2\right )+\frac {1}{2} a^2 d (7 c+d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{4 a^2 \left (c^2-d^2\right )^2}\\ &=\frac {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 {d (7 c+d) \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 {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{(c-d)^3}-\frac {\left (d \left (15 c^2+10 c d+7 d^2\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 {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 {d (7 c+d) \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 \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 (d \left (15 c^2+10 c d+7 d^2\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} \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 {\sqrt {d} \left (15 c^2+10 c d+7 d^2\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 (c+d)^{5/2} f}+\frac {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 {d (7 c+d) \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 = 4.98, size = 414, normalized size = 1.68 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {\sqrt {d} \left (15 c^2+10 c d+7 d^2\right ) \left (2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (\sqrt {c+d}-\sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{(c-d)^3 (c+d)^{5/2}}+\frac {\sqrt {d} \left (15 c^2+10 c d+7 d^2\right ) \left (2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (\sqrt {c+d}+\sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )-\sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{(d-c)^3 (c+d)^{5/2}}+\frac {4 d (7 c+d) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d)^2 (c+d)^2 (c+d \sin (e+f x))}+\frac {8 d \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d) (c+d) (c+d \sin (e+f x))^2}+\frac {(32+32 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )}{(c-d)^3}\right )}{16 f \sqrt {a (\sin (e+f x)+1)}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(((32 + 32*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e +

f*x)/4])])/(c - d)^3 + (Sqrt[d]*(15*c^2 + 10*c*d + 7*d^2)*(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])]))/((c - d)^3*(c + d)^(5/2)) +

(Sqrt[d]*(15*c^2 + 10*c*d + 7*d^2)*(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])]))/((-c + d)^3*(c + d)^(5/2)) + (8*d*(Cos[(e + f*x)/

2] - Sin[(e + f*x)/2]))/((c - d)*(c + d)*(c + d*Sin[e + f*x])^2) + (4*d*(7*c + d)*(Cos[(e + f*x)/2] - Sin[(e +

f*x)/2]))/((c - d)^2*(c + d)^2*(c + d*Sin[e + f*x]))))/(16*f*Sqrt[a*(1 + Sin[e + f*x])])

________________________________________________________________________________________

fricas [B] time = 1.20, size = 2903, normalized size = 11.75 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/16*((15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*co

s(f*x + e)^3 - (30*a*c^3*d + 35*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + (15*a*c^4 + 10*a*c^3*d + 22

*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e) + (15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4

- (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + 2*(15*a*c^3*d + 10*a*c^2*d^2 + 7*a*c*d^3)*cos(f*x +

e))*sin(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*co

s(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*c^4 + 4*a*c^3*d + 6*a*c^2

*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d

^3 + a*d^4)*cos(f*x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a

*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c

^2*d^2 + a*c*d^3)*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*(9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 + (7*

c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e)^2 + (9*c^3*d - 8*c^2*d^2 - 3*c*d^3 + 2*d^4)*cos(f*x + e) - (9*c^3*d - 15

*c^2*d^2 + 3*c*d^3 + 3*d^4 - (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))

/((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^3 + (2*a*c^6*d - a*c^5*

d^2 - 5*a*c^4*d^3 + 2*a*c^3*d^4 + 4*a*c^2*d^5 - a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - (a*c^7 - a*c^6*d - a*c^5*d

^2 + a*c^4*d^3 - a*c^3*d^4 + a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*

a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f + ((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d

^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - 2*(a*c^6*d - a*c^5*d^2 - 2*a*c^4*d^3 + 2*a*c^3*d^4 + a*c^2*d^5 - a*c*

d^6)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d

^7)*f)*sin(f*x + e)), -1/8*((15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*

a*c*d^3 + 7*a*d^4)*cos(f*x + e)^3 - (30*a*c^3*d + 35*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + (15*a*

c^4 + 10*a*c^3*d + 22*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e) + (15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 +

24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + 2*(15*a*c^3*d + 10*a*c^2*d^2 +

7*a*c*d^3)*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)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*sqrt(2)*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 +

a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*

x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d

^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c^2*d^2 + a*c*d^3)*

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*(9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 + (7*c^2*d^2 - 6*c*d^3

- d^4)*cos(f*x + e)^2 + (9*c^3*d - 8*c^2*d^2 - 3*c*d^3 + 2*d^4)*cos(f*x + e) - (9*c^3*d - 15*c^2*d^2 + 3*c*d^3

+ 3*d^4 - (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^5*d^2 - a*c

^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^3 + (2*a*c^6*d - a*c^5*d^2 - 5*a*c^4*d^3

+ 2*a*c^3*d^4 + 4*a*c^2*d^5 - a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - (a*c^7 - a*c^6*d - a*c^5*d^2 + a*c^4*d^3 - a

*c^3*d^4 + a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^

3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f + ((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d

^7)*f*cos(f*x + e)^2 - 2*(a*c^6*d - a*c^5*d^2 - 2*a*c^4*d^3 + 2*a*c^3*d^4 + a*c^2*d^5 - a*c*d^6)*f*cos(f*x + e

) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f)*sin(f*x + e

))]

________________________________________________________________________________________

giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^(1/2)/(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,x):;OUTPUT:Unab

le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (4*p

i/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unab

le to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*p

i/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unab

le to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*p

i/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign

: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Warning, integration of abs or sign assumes constant sign by intervals (

correct if the argument is real):Check [abs(cos((f*t_nostep+exp(1))/2-pi/4))]Unable to check sign: (4*pi/t_nos

tep/2)>(-4*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Discontinuities at zeroes o

f cos((f*t_nostep+exp(1))/2-pi/4) were not checkedUnable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Un

able to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_noste

p/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t

_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/