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

Optimal. Leaf size=154 \[ \frac{2 b^2 B}{a d \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}-\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.493671, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {21, 3569, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{2 b^2 B}{a d \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}-\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

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]

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 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 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

Mathematica [A]  time = 1.15979, size = 166, normalized size = 1.08 \[ \frac{B \left (-\frac{2 \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{2 b^2}{\sqrt{a+b \tan (c+d x)}}+\frac{a (a+i b) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}+\frac{a (a-i b) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}\right )}{a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 1.17, size = 39987, normalized size = 259.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.60152, size = 29088, normalized size = 188.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^
11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*
a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^
4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b
^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4
*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(-((3*B^6*a^
12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B
^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b
^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*
b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*
b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8
- 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2
- 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^
4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^
4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 +
b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8
)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9
*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8
)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a
^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 -
8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4
+ B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4
))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(
d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*((3*B^3*a^16 + 14*B^3*
a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^
14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B
^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 +
14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B
^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^
8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a
*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a
^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + 4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2
*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x
+ c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^
2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*
B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4))*arctan(((3*B^6*a^12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6
*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*
b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*
a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6
*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((a^14 + 5*a^12*b^2 + 9*
a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15
*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*
b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6
*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 -
6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*
B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(
B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5
*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^
5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^
4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))
/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b
- 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)
- sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10
- 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9
*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^1
0 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*
b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9
- 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6
*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(
d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a
^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^
4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 -
8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 +
B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*
b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*
((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt
((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*s
in(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2
+ B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((B^
4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2
+ B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b
- 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6
)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^
2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^
2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 +
12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(
B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^
5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6
- 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))
*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) +
(9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/c
os(d*x + c)) + 2*(B^5*a^2*b^2 + B^5*b^4 + (B^5*a^4 - B^5*b^4)*cos(d*x + c)^2 + 2*(B^5*a^3*b + B^5*a*b^3)*cos(d
*x + c)*sin(d*x + c))*sqrt(a)*log(-(8*a*b*cos(d*x + c)*sin(d*x + c) + (8*a^2 - b^2)*cos(d*x + c)^2 + b^2 - 4*(
2*a*cos(d*x + c)^2 + b*cos(d*x + c)*sin(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))
)/(cos(d*x + c)^2 - 1)) + 8*(B^5*a^2*b^2*cos(d*x + c)^2 + B^5*a*b^3*cos(d*x + c)*sin(d*x + c))*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c)))/((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)
*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d), 1/4*(4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4
- 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b
^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a
^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*
b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(-((3*B^6*a^12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a
^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sq
rt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^
2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4
*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sq
rt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4
+ 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*
a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*
B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^
4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^
9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6
+ 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c
))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^
4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20
*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^
6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6
*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3
*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B
^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + 4*sqrt(2)
*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^
3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4
*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 -
3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*
(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(((3*B^6*a^12 + 14*B^6*a^10
*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4
+ 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)
*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 -
5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4
+ B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11
+ 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^
4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sq
rt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^
4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*c
os(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6
)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a
*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt
((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^
6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*
x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) - sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^
3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d
^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2
+ 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B
^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*s
qrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) +
b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*
a^2*b^4 + B^10*b^6)) + sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d
*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6
)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*
b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 +
3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4
*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 -
15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*
b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*
a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6
*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a
^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 -
B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x +
c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*
d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 +
B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*
a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*c
os(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4
*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x
+ c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x
+ c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) + 8*(B^5*a^2*b^2 + B^5*b^4 + (B^5*a
^4 - B^5*b^4)*cos(d*x + c)^2 + 2*(B^5*a^3*b + B^5*a*b^3)*cos(d*x + c)*sin(d*x + c))*sqrt(-a)*arctan(sqrt(-a)*s
qrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))/a) + 8*(B^5*a^2*b^2*cos(d*x + c)^2 + B^5*a*b^3*cos(d*x + c
)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^
2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} B \int \frac{\cot{\left (c + d x \right )}}{a \sqrt{a + b \tan{\left (c + d x \right )}} + b \sqrt{a + b \tan{\left (c + d x \right )}} \tan{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**(5/2),x)

[Out]

B*Integral(cot(c + d*x)/(a*sqrt(a + b*tan(c + d*x)) + b*sqrt(a + b*tan(c + d*x))*tan(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B b \tan \left (d x + c\right ) + B a\right )} \cot \left (d x + c\right )}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((B*b*tan(d*x + c) + B*a)*cot(d*x + c)/(b*tan(d*x + c) + a)^(5/2), x)