∫ −a+b tan(c+dx)_ (a+b tan(c+dx))5∕2 dx

3.372 \(\int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \]

[Out]

(I*a-b)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-(I*a+b)*arctanh((a+b*tan(d*x+c))^(1/2)/(

a+I*b)^(1/2))/(a+I*b)^(5/2)/d+2*b*(3*a^2-b^2)/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)+4/3*a*b/(a^2+b^2)/d/(a+b*ta

n(d*x+c))^(3/2)

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Rubi [A] time = 0.34, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3529, 3539, 3537, 63, 208} \[ \frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((I*a - b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*a + b)*ArcTanh[Sqrt[a +

b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (4*a*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (

2*b*(3*a^2 - b^2))/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

b, c, d, m, n, x]

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 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((

b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +

f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c

- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*

d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&

NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c

+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]

)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]

&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {\int \frac {-a^2+b^2+2 a b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx}{a^2+b^2}\\ &=\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {-a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {(a-i b) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac {(a+i b) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}\\ &=\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {(i a-b) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac {(i a+b) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {(a+i b) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b (i a+b)^2 d}-\frac {(i (i a+b)) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a+i b)^2 b d}\\ &=\frac {(i a-b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(i a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [C] time = 0.35, size = 156, normalized size = 0.90 \[ -\frac {i \cos (c+d x) (a-b \tan (c+d x)) \left ((a+i b)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{3 d (a-i b) (a+i b) (a+b \tan (c+d x))^{3/2} (a \cos (c+d x)-b \sin (c+d x))} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-1/3*I)*Cos[c + d*x]*((a + I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*

b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a - b*Tan[c + d*x]))/((a - I*b)*(a + I

*b)*d*(a*Cos[c + d*x] - b*Sin[c + d*x])*(a + b*Tan[c + d*x])^(3/2))

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fricas [B] time = 1.77, size = 10036, normalized size = 57.68 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+b*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(2)*((a^14 - a^12*b^2 - 19*a^10*b^4 - 45*a^8*b^6 - 45*a^6*b^8 - 19*a^4*b^10 - a^2*b^12 + b^14)*d^

5*cos(d*x + c)^4 + 2*(3*a^12*b^2 + 14*a^10*b^4 + 25*a^8*b^6 + 20*a^6*b^8 + 5*a^4*b^10 - 2*a^2*b^12 - b^14)*d^5

*cos(d*x + c)^2 + (a^10*b^4 + 5*a^8*b^6 + 10*a^6*b^8 + 10*a^4*b^10 + 5*a^2*b^12 + b^14)*d^5 + 4*((a^13*b + 4*a

^11*b^3 + 5*a^9*b^5 - 5*a^5*b^9 - 4*a^3*b^11 - a*b^13)*d^5*cos(d*x + c)^3 + (a^11*b^3 + 5*a^9*b^5 + 10*a^7*b^7

+ 10*a^5*b^9 + 5*a^3*b^11 + a*b^13)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 3

5*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 1

10*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49

*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((49*a^12*b^2

- 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a

^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^1

8 + b^20)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(-((7*a^20 + 14*a^18*b^2 - 77*a^16*b

^4 - 344*a^14*b^6 - 546*a^12*b^8 - 364*a^10*b^10 + 14*a^8*b^12 + 168*a^6*b^14 + 91*a^4*b^16 + 14*a^2*b^18 - b^

20)*d^4*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((

a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 +

45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (7*a^17 - 84*a^13

*b^4 - 176*a^11*b^6 - 110*a^9*b^8 + 32*a^7*b^10 + 60*a^5*b^12 + 16*a^3*b^14 - a*b^16)*d^2*sqrt((49*a^12*b^2 -

490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16

*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 +

b^20)*d^4)) + sqrt(2)*(4*(a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 -

a*b^14)*d^7*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14

)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^

14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^12 + 14*

a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 151

9*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^

6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt

((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^1

5*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*

a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 4

2*a^2*b^12 + b^14))*sqrt(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a

^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*

a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((147*a^20*b^3 - 1078*a^18*b^5 + 931*a^16*b^7 + 4760*a^14*b^9 - 12

74*a^12*b^11 - 4452*a^10*b^13 + 1214*a^8*b^15 + 1240*a^6*b^17 - 505*a^4*b^19 + 42*a^2*b^21 - b^23)*d^3*sqrt(1/

((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 4*(49*a^17*b^3 - 490*a^15*b^5 + 1470*a^13*b^7 - 994*

a^11*b^9 - 1008*a^9*b^11 + 1442*a^7*b^13 - 510*a^5*b^15 + 42*a^3*b^17 - a*b^19)*d*cos(d*x + c))*sqrt((a^14 + 7

*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60

*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 +

3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12

+ b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(

1/4) + (49*a^17*b^2 - 392*a^15*b^4 + 588*a^13*b^6 + 1064*a^11*b^8 - 938*a^9*b^10 - 504*a^7*b^12 + 428*a^5*b^14

- 40*a^3*b^16 + a*b^18)*cos(d*x + c) + (49*a^16*b^3 - 392*a^14*b^5 + 588*a^12*b^7 + 1064*a^10*b^9 - 938*a^8*b

^11 - 504*a^6*b^13 + 428*a^4*b^15 - 40*a^2*b^17 + b^19)*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a

^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*(4*(7*a^23*b + 7*a^21*b^3 - 91*a^19*b^5 - 267*a^17*b^7 - 202*a^15*b^9 + 18

2*a^13*b^11 + 378*a^11*b^13 + 154*a^9*b^15 - 77*a^7*b^17 - 77*a^5*b^19 - 15*a^3*b^21 + a*b^23)*d^7*sqrt((49*a^

12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2

+ 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a

^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (21*a^20*b + 14*a^18*b^3 - 259*a^16*

b^5 - 696*a^14*b^7 - 598*a^12*b^9 + 52*a^10*b^11 + 354*a^8*b^13 + 136*a^6*b^15 - 31*a^4*b^17 - 18*a^2*b^19 + b

^21)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/(

(a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14

+ 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 2

1*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10

+ 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 +

1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(

d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 14

84*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)) + 12*sqrt(2)*((a^14 - a^12*b^2 - 19*a^10*b^4 - 45*a^8*b^6 - 4

5*a^6*b^8 - 19*a^4*b^10 - a^2*b^12 + b^14)*d^5*cos(d*x + c)^4 + 2*(3*a^12*b^2 + 14*a^10*b^4 + 25*a^8*b^6 + 20*

a^6*b^8 + 5*a^4*b^10 - 2*a^2*b^12 - b^14)*d^5*cos(d*x + c)^2 + (a^10*b^4 + 5*a^8*b^6 + 10*a^6*b^8 + 10*a^4*b^1

0 + 5*a^2*b^12 + b^14)*d^5 + 4*((a^13*b + 4*a^11*b^3 + 5*a^9*b^5 - 5*a^5*b^9 - 4*a^3*b^11 - a*b^13)*d^5*cos(d*

x + c)^3 + (a^11*b^3 + 5*a^9*b^5 + 10*a^7*b^7 + 10*a^5*b^9 + 5*a^3*b^11 + a*b^13)*d^5*cos(d*x + c))*sin(d*x +

c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17

- 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((

a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*

b^10 - 42*a^2*b^12 + b^14))*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42

*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b

^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/

4)*arctan(((7*a^20 + 14*a^18*b^2 - 77*a^16*b^4 - 344*a^14*b^6 - 546*a^12*b^8 - 364*a^10*b^10 + 14*a^8*b^12 + 1

68*a^6*b^14 + 91*a^4*b^16 + 14*a^2*b^18 - b^20)*d^4*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6

*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 2

52*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2

+ 3*a^2*b^4 + b^6)*d^4)) + (7*a^17 - 84*a^13*b^4 - 176*a^11*b^6 - 110*a^9*b^8 + 32*a^7*b^10 + 60*a^5*b^12 + 16

*a^3*b^14 - a*b^16)*d^2*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2

*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12

+ 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*(4*(a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*

b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*d^7*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484

*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8

+ 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*

b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12

)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^

20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 4

5*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a

^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 8

4*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 15

19*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b

^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^

20 + b^22)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((147*a^20*b^3 - 1078*

a^18*b^5 + 931*a^16*b^7 + 4760*a^14*b^9 - 1274*a^12*b^11 - 4452*a^10*b^13 + 1214*a^8*b^15 + 1240*a^6*b^17 - 50

5*a^4*b^19 + 42*a^2*b^21 - b^23)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 4*(49*a^

17*b^3 - 490*a^15*b^5 + 1470*a^13*b^7 - 994*a^11*b^9 - 1008*a^9*b^11 + 1442*a^7*b^13 - 510*a^5*b^15 + 42*a^3*b

^17 - a*b^19)*d*cos(d*x + c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 +

7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12

- 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6

- 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/

((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (49*a^17*b^2 - 392*a^15*b^4 + 588*a^13*b^6 + 1064*a^11*b^8

- 938*a^9*b^10 - 504*a^7*b^12 + 428*a^5*b^14 - 40*a^3*b^16 + a*b^18)*cos(d*x + c) + (49*a^16*b^3 - 392*a^14*b^

5 + 588*a^12*b^7 + 1064*a^10*b^9 - 938*a^8*b^11 - 504*a^6*b^13 + 428*a^4*b^15 - 40*a^2*b^17 + b^19)*sin(d*x +

c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) - sqrt(2)*(4*(7*a^23*b + 7*a^21*b^3 - 91

*a^19*b^5 - 267*a^17*b^7 - 202*a^15*b^9 + 182*a^13*b^11 + 378*a^11*b^13 + 154*a^9*b^15 - 77*a^7*b^17 - 77*a^5*

b^19 - 15*a^3*b^21 + a*b^23)*d^7*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10

- 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*

a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*

d^4)) + (21*a^20*b + 14*a^18*b^3 - 259*a^16*b^5 - 696*a^14*b^7 - 598*a^12*b^9 + 52*a^10*b^11 + 354*a^8*b^13 +

136*a^6*b^15 - 31*a^4*b^17 - 18*a^2*b^19 + b^21)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^

6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 +

252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^14 + 7*a^12*b^2

+ 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4

- 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4

+ b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))

*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(49

*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)) - 3*sqrt(2)*((a^8

- 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*d*cos(d*x + c)^4 + 2*(3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d*co

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

b^3 + 2*a^3*b^5 + a*b^7)*d*cos(d*x + c))*sin(d*x + c) - ((a^11 - 27*a^9*b^2 + 162*a^7*b^4 - 238*a^5*b^6 + 77*a

^3*b^8 - 7*a*b^10)*d^3*cos(d*x + c)^4 + 2*(3*a^9*b^2 - 64*a^7*b^4 + 126*a^5*b^6 - 56*a^3*b^8 + 7*a*b^10)*d^3*c

os(d*x + c)^2 + (a^7*b^4 - 21*a^5*b^6 + 35*a^3*b^8 - 7*a*b^10)*d^3 + 4*((a^10*b - 22*a^8*b^3 + 56*a^6*b^5 - 42

*a^4*b^7 + 7*a^2*b^9)*d^3*cos(d*x + c)^3 + (a^8*b^3 - 21*a^6*b^5 + 35*a^4*b^7 - 7*a^2*b^9)*d^3*cos(d*x + c))*s

in(d*x + c))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8

*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^

9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12

*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*(1/((a^6 + 3*a^4*b^2 +

3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10

- 1316*a^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt(1/((a^6 + 3*a^4

*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((147*a^20*b^3 - 1078*a^18*b^5 + 931*a^16*b^7 + 4760*a^14

*b^9 - 1274*a^12*b^11 - 4452*a^10*b^13 + 1214*a^8*b^15 + 1240*a^6*b^17 - 505*a^4*b^19 + 42*a^2*b^21 - b^23)*d^

3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 4*(49*a^17*b^3 - 490*a^15*b^5 + 1470*a^13*b

^7 - 994*a^11*b^9 - 1008*a^9*b^11 + 1442*a^7*b^13 - 510*a^5*b^15 + 42*a^3*b^17 - a*b^19)*d*cos(d*x + c))*sqrt(

(a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^17 - 16*a^15

*b^2 - 60*a^13*b^4 - 32*a^11*b^6 + 110*a^9*b^8 + 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d^2*sqrt(1/((a^6 + 3*a

^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42

*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6

)*d^4))^(1/4) + (49*a^17*b^2 - 392*a^15*b^4 + 588*a^13*b^6 + 1064*a^11*b^8 - 938*a^9*b^10 - 504*a^7*b^12 + 428

*a^5*b^14 - 40*a^3*b^16 + a*b^18)*cos(d*x + c) + (49*a^16*b^3 - 392*a^14*b^5 + 588*a^12*b^7 + 1064*a^10*b^9 -

938*a^8*b^11 - 504*a^6*b^13 + 428*a^4*b^15 - 40*a^2*b^17 + b^19)*sin(d*x + c))/cos(d*x + c)) + 3*sqrt(2)*((a^8