∫ A+B sin(e+fx)_______ (a+a sin(e+fx))5∕2(c+d sin(e+fx))2 dx

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

Optimal. Leaf size=395 \[ -\frac {\left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^4}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {(3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))} \]

[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)

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Rubi [A] time = 1.54, antiderivative size = 395, normalized size of antiderivative = 1.00, 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} \[ -\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac {\left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^4}-\frac {(3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((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]*Sqrt[a

+ a*Sin[e + f*x]])])/(16*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*S

in[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 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]

Rubi steps

\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx &=-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\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 ) \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 )}{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 ) \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 )}{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 ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^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 ) \tanh ^{-1}\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*}

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Mathematica [C] time = 12.37, size = 1318, normalized size = 3.34 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

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

] + Sin[(e + f*x)/2])^5)/(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] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d]

- Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(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*Cos[(e + f*x)/2] + 8*B*d^3*Cos[(e + f*x)/2] - 6*A*c^3*C

os[(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*Si

n[(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*Sin[(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*Si

n[(5*(e + f*x))/2] + 16*A*c*d^2*Sin[(5*(e + f*x))/2] - 32*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]))

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fricas [B] time = 14.83, size = 5151, normalized size = 13.04 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/64*(sqrt(2)*(4*(3*A + 5*B)*c^4 - 64*(A + 3*B)*c^3*d + 8*(37*A - 77*B)*c^2*d^2 + 64*(13*A - 9*B)*c*d^3 + 4*

(115*A - 43*B)*d^4 + ((3*A + 5*B)*c^3*d - (19*A + 53*B)*c^2*d^2 + (93*A - 101*B)*c*d^3 + (115*A - 43*B)*d^4)*c

os(f*x + e)^4 - ((3*A + 5*B)*c^4 - (13*A + 43*B)*c^3*d + (55*A - 207*B)*c^2*d^2 + 7*(43*A - 35*B)*c*d^3 + 2*(1

15*A - 43*B)*d^4)*cos(f*x + e)^3 - (3*(3*A + 5*B)*c^4 - 2*(21*A + 67*B)*c^3*d + 8*(23*A - 71*B)*c^2*d^2 + 2*(4

05*A - 317*B)*c*d^3 + 5*(115*A - 43*B)*d^4)*cos(f*x + e)^2 + 2*((3*A + 5*B)*c^4 - 16*(A + 3*B)*c^3*d + 2*(37*A

- 77*B)*c^2*d^2 + 16*(13*A - 9*B)*c*d^3 + (115*A - 43*B)*d^4)*cos(f*x + e) + (4*(3*A + 5*B)*c^4 - 64*(A + 3*B

)*c^3*d + 8*(37*A - 77*B)*c^2*d^2 + 64*(13*A - 9*B)*c*d^3 + 4*(115*A - 43*B)*d^4 - ((3*A + 5*B)*c^3*d - (19*A

+ 53*B)*c^2*d^2 + (93*A - 101*B)*c*d^3 + (115*A - 43*B)*d^4)*cos(f*x + e)^3 - ((3*A + 5*B)*c^4 - 2*(5*A + 19*B

)*c^3*d + 4*(9*A - 65*B)*c^2*d^2 + 2*(197*A - 173*B)*c*d^3 + 3*(115*A - 43*B)*d^4)*cos(f*x + e)^2 + 2*((3*A +

5*B)*c^4 - 16*(A + 3*B)*c^3*d + 2*(37*A - 77*B)*c^2*d^2 + 16*(13*A - 9*B)*c*d^3 + (115*A - 43*B)*d^4)*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)) - 16*(20*B*a*c^3*d - 4*(7*A - 10*B)*a*c^2*d^2 - 4*(12*A - 7*B)*a*

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

a*c^3*d - (7*A - 15*B)*a*c^2*d^2 - (19*A - 12*B)*a*c*d^3 - 2*(5*A - 2*B)*a*d^4)*cos(f*x + e)^3 - (15*B*a*c^3*d

- (21*A - 40*B)*a*c^2*d^2 - (50*A - 31*B)*a*c*d^3 - 5*(5*A - 2*B)*a*d^4)*cos(f*x + e)^2 + 2*(5*B*a*c^3*d - (7

*A - 10*B)*a*c^2*d^2 - (12*A - 7*B)*a*c*d^3 - (5*A - 2*B)*a*d^4)*cos(f*x + e) + (20*B*a*c^3*d - 4*(7*A - 10*B)

*a*c^2*d^2 - 4*(12*A - 7*B)*a*c*d^3 - 4*(5*A - 2*B)*a*d^4 - (5*B*a*c^2*d^2 - (7*A - 5*B)*a*c*d^3 - (5*A - 2*B)

*a*d^4)*cos(f*x + e)^3 - (5*B*a*c^3*d - (7*A - 20*B)*a*c^2*d^2 - (26*A - 17*B)*a*c*d^3 - 3*(5*A - 2*B)*a*d^4)*

cos(f*x + e)^2 + 2*(5*B*a*c^3*d - (7*A - 10*B)*a*c^2*d^2 - (12*A - 7*B)*a*c*d^3 - (5*A - 2*B)*a*d^4)*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*c

os(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(4*(A - B)*c^4 - 8*(A - B)*c^3*d +

8*(A - B)*c*d^3 - 4*(A - B)*d^4 - ((3*A + 5*B)*c^3*d - (19*A - 27*B)*c^2*d^2 - (19*A + 21*B)*c*d^3 + (35*A - 1

1*B)*d^4)*cos(f*x + e)^3 + ((3*A + 5*B)*c^4 - (15*A - 7*B)*c^3*d - (7*A - 15*B)*c^2*d^2 - (A + 23*B)*c*d^3 + 4

*(5*A - B)*d^4)*cos(f*x + e)^2 + ((7*A + B)*c^4 - 20*(A - B)*c^3*d - 2*(13*A - 21*B)*c^2*d^2 - 4*(3*A + 13*B)*

c*d^3 + (51*A - 11*B)*d^4)*cos(f*x + e) - (4*(A - B)*c^4 - 8*(A - B)*c^3*d + 8*(A - B)*c*d^3 - 4*(A - B)*d^4 -

((3*A + 5*B)*c^3*d - (19*A - 27*B)*c^2*d^2 - (19*A + 21*B)*c*d^3 + (35*A - 11*B)*d^4)*cos(f*x + e)^2 - ((3*A

+ 5*B)*c^4 - 12*(A - B)*c^3*d - 2*(13*A - 21*B)*c^2*d^2 - 4*(5*A + 11*B)*c*d^3 + 5*(11*A - 3*B)*d^4)*cos(f*x +

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

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

a^3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*

d^4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^

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

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

d^5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a

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

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

+ a^3*d^6)*f)*sin(f*x + e)), -1/64*(sqrt(2)*(4*(3*A + 5*B)*c^4 - 64*(A + 3*B)*c^3*d + 8*(37*A - 77*B)*c^2*d^2

+ 64*(13*A - 9*B)*c*d^3 + 4*(115*A - 43*B)*d^4 + ((3*A + 5*B)*c^3*d - (19*A + 53*B)*c^2*d^2 + (93*A - 101*B)*

c*d^3 + (115*A - 43*B)*d^4)*cos(f*x + e)^4 - ((3*A + 5*B)*c^4 - (13*A + 43*B)*c^3*d + (55*A - 207*B)*c^2*d^2 +

7*(43*A - 35*B)*c*d^3 + 2*(115*A - 43*B)*d^4)*cos(f*x + e)^3 - (3*(3*A + 5*B)*c^4 - 2*(21*A + 67*B)*c^3*d + 8

*(23*A - 71*B)*c^2*d^2 + 2*(405*A - 317*B)*c*d^3 + 5*(115*A - 43*B)*d^4)*cos(f*x + e)^2 + 2*((3*A + 5*B)*c^4 -

16*(A + 3*B)*c^3*d + 2*(37*A - 77*B)*c^2*d^2 + 16*(13*A - 9*B)*c*d^3 + (115*A - 43*B)*d^4)*cos(f*x + e) + (4*

(3*A + 5*B)*c^4 - 64*(A + 3*B)*c^3*d + 8*(37*A - 77*B)*c^2*d^2 + 64*(13*A - 9*B)*c*d^3 + 4*(115*A - 43*B)*d^4

- ((3*A + 5*B)*c^3*d - (19*A + 53*B)*c^2*d^2 + (93*A - 101*B)*c*d^3 + (115*A - 43*B)*d^4)*cos(f*x + e)^3 - ((3

*A + 5*B)*c^4 - 2*(5*A + 19*B)*c^3*d + 4*(9*A - 65*B)*c^2*d^2 + 2*(197*A - 173*B)*c*d^3 + 3*(115*A - 43*B)*d^4

)*cos(f*x + e)^2 + 2*((3*A + 5*B)*c^4 - 16*(A + 3*B)*c^3*d + 2*(37*A - 77*B)*c^2*d^2 + 16*(13*A - 9*B)*c*d^3 +

(115*A - 43*B)*d^4)*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)) + 32*(20*B*a*c^3*d - 4*(7*A - 10*B)*

a*c^2*d^2 - 4*(12*A - 7*B)*a*c*d^3 - 4*(5*A - 2*B)*a*d^4 + (5*B*a*c^2*d^2 - (7*A - 5*B)*a*c*d^3 - (5*A - 2*B)*

a*d^4)*cos(f*x + e)^4 - (5*B*a*c^3*d - (7*A - 15*B)*a*c^2*d^2 - (19*A - 12*B)*a*c*d^3 - 2*(5*A - 2*B)*a*d^4)*c

os(f*x + e)^3 - (15*B*a*c^3*d - (21*A - 40*B)*a*c^2*d^2 - (50*A - 31*B)*a*c*d^3 - 5*(5*A - 2*B)*a*d^4)*cos(f*x

+ e)^2 + 2*(5*B*a*c^3*d - (7*A - 10*B)*a*c^2*d^2 - (12*A - 7*B)*a*c*d^3 - (5*A - 2*B)*a*d^4)*cos(f*x + e) + (

20*B*a*c^3*d - 4*(7*A - 10*B)*a*c^2*d^2 - 4*(12*A - 7*B)*a*c*d^3 - 4*(5*A - 2*B)*a*d^4 - (5*B*a*c^2*d^2 - (7*A

- 5*B)*a*c*d^3 - (5*A - 2*B)*a*d^4)*cos(f*x + e)^3 - (5*B*a*c^3*d - (7*A - 20*B)*a*c^2*d^2 - (26*A - 17*B)*a*

c*d^3 - 3*(5*A - 2*B)*a*d^4)*cos(f*x + e)^2 + 2*(5*B*a*c^3*d - (7*A - 10*B)*a*c^2*d^2 - (12*A - 7*B)*a*c*d^3 -

(5*A - 2*B)*a*d^4)*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*si

n(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + 4*(4*(A - B)*c^4 - 8*(A - B)*c^3*d + 8*(A - B)*

c*d^3 - 4*(A - B)*d^4 - ((3*A + 5*B)*c^3*d - (19*A - 27*B)*c^2*d^2 - (19*A + 21*B)*c*d^3 + (35*A - 11*B)*d^4)*

cos(f*x + e)^3 + ((3*A + 5*B)*c^4 - (15*A - 7*B)*c^3*d - (7*A - 15*B)*c^2*d^2 - (A + 23*B)*c*d^3 + 4*(5*A - B)

*d^4)*cos(f*x + e)^2 + ((7*A + B)*c^4 - 20*(A - B)*c^3*d - 2*(13*A - 21*B)*c^2*d^2 - 4*(3*A + 13*B)*c*d^3 + (5

1*A - 11*B)*d^4)*cos(f*x + e) - (4*(A - B)*c^4 - 8*(A - B)*c^3*d + 8*(A - B)*c*d^3 - 4*(A - B)*d^4 - ((3*A + 5

*B)*c^3*d - (19*A - 27*B)*c^2*d^2 - (19*A + 21*B)*c*d^3 + (35*A - 11*B)*d^4)*cos(f*x + e)^2 - ((3*A + 5*B)*c^4

- 12*(A - B)*c^3*d - 2*(13*A - 21*B)*c^2*d^2 - 4*(5*A + 11*B)*c*d^3 + 5*(11*A - 3*B)*d^4)*cos(f*x + e))*sin(f

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

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

+ 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a

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

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

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

d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*

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

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

)*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((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]

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: (2*p

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

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

eck 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)Warnin

g, 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_nostep/2)>(-4*pi/t_nostep/2)Unable to check sig

n: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Discontinuities at zeroes of cos((f*t_nostep+exp(1))/2-pi/4) were not c

heckedUnable 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/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_nost

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