3.339 \(\int \frac {\cot ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=236 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^4}{a d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a-b)^2 (\sec (c+d x)+1)}+\frac {(4 a-7 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{5/2}}+\frac {(4 a+7 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{5/2}} \]

[Out]

-2*arctanh((a+b*sec(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+1/4*(4*a-7*b)*arctanh((a+b*sec(d*x+c))^(1/2)/(a-b)^(1/2))
/(a-b)^(5/2)/d+1/4*(4*a+7*b)*arctanh((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2))/(a+b)^(5/2)/d+2*b^4/a/(a^2-b^2)^2/d/(
a+b*sec(d*x+c))^(1/2)+1/4*(a+b*sec(d*x+c))^(1/2)/(a+b)^2/d/(1-sec(d*x+c))+1/4*(a+b*sec(d*x+c))^(1/2)/(a-b)^2/d
/(1+sec(d*x+c))

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Rubi [A]  time = 0.38, antiderivative size = 316, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3885, 898, 1335, 206, 199} \[ \frac {2 b^4}{a d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{5/2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{5/2}}+\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{2 d (a-b)^{5/2}}+\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{2 d (a+b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqr
t[a - b]])/(2*(a - b)^(5/2)*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + (b*Ar
cTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x
]]/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) + (2*b^4)/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + Sqrt[a + b*Sec[c
+ d*x]]/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)^2*d*(1 + Sec[c + d*x]))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 898

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1335

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x]
&& NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [B]  time = 7.53, size = 1114, normalized size = 4.72 \[ \frac {(b+a \cos (c+d x))^2 \left (-\frac {2 b^5}{a^2 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}+\frac {\left (-a^2+2 b \cos (c+d x) a-b^2\right ) \csc ^2(c+d x)}{2 \left (b^2-a^2\right )^2}+\frac {a^4+b^2 a^2+4 b^4}{2 a^2 \left (b^2-a^2\right )^2}\right ) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}-\frac {(b+a \cos (c+d x))^{3/2} \left (-\frac {\left (2 a^4-6 b^2 a^2-2 b^4\right ) \sqrt {-a \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\sqrt {a-b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a-b} \sqrt {-a \cos (c+d x)}}\right )+(a-b) \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a+b} \sqrt {-a \cos (c+d x)}}\right )\right )}{\sqrt {a} (a-b) (a+b)}-\frac {a \left (2 a^4-4 b^2 a^2+2 b^4\right ) \left (4 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b+a \cos (c+d x)}}{\sqrt {-a \cos (c+d x)}}\right )-\sqrt {a} \left (\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a-b} \sqrt {-a \cos (c+d x)}}\right )+\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a+b} \sqrt {-a \cos (c+d x)}}\right )\right )\right ) \sqrt {-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt {\sec (c+d x)}}{\sqrt {a-b} \sqrt {a+b} \left (a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right )}-\frac {a \left (7 a b^3-a^3 b\right ) \left (-\sqrt {-a^2} \sqrt {a+b} \log \left (\sqrt {b+a \cos (c+d x)}-\sqrt {b-a}\right )+\sqrt {-a^2} \sqrt {a+b} \log \left (\sqrt {b-a}+\sqrt {b+a \cos (c+d x)}\right )-a \sqrt {b-a} \log \left (\sqrt {b+a \cos (c+d x)}-\sqrt {a+b}\right )+a \sqrt {b-a} \log \left (\sqrt {a+b}+\sqrt {b+a \cos (c+d x)}\right )+\sqrt {-a^2} \sqrt {a+b} \log \left (b+\sqrt {a} \sqrt {-a \cos (c+d x)}-\sqrt {b-a} \sqrt {b+a \cos (c+d x)}\right )-\sqrt {-a^2} \sqrt {a+b} \log \left (b+\sqrt {a} \sqrt {-a \cos (c+d x)}+\sqrt {b-a} \sqrt {b+a \cos (c+d x)}\right )+a \sqrt {b-a} \log \left (b+\sqrt {-a} \sqrt {-a \cos (c+d x)}-\sqrt {a+b} \sqrt {b+a \cos (c+d x)}\right )-a \sqrt {b-a} \log \left (b+\sqrt {-a} \sqrt {-a \cos (c+d x)}+\sqrt {a+b} \sqrt {b+a \cos (c+d x)}\right )\right )}{2 (-a)^{3/2} \sqrt {b-a} \sqrt {a+b} \sqrt {-a \cos (c+d x)} \sqrt {\sec (c+d x)}}\right ) \sec ^{\frac {3}{2}}(c+d x)}{4 a (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

-1/4*((b + a*Cos[c + d*x])^(3/2)*(-1/2*(a*(-(a^3*b) + 7*a*b^3)*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + S
qrt[b + a*Cos[c + d*x]]]) + Sqrt[-a^2]*Sqrt[a + b]*Log[Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a +
b]*Log[-Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] +
Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] - Sqr
t[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[
-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[
b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]]))/((-a)^(3/2)*Sqrt[-a + b]*Sqrt[a
+ b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) - ((2*a^4 - 6*a^2*b^2 - 2*b^4)*(Sqrt[a - b]*(a + b)*ArcTan[(
Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + (a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a]
*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])
/(Sqrt[a]*(a - b)*(a + b)) - (a*(2*a^4 - 4*a^2*b^2 + 2*b^4)*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c
+ d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b
]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[
c + d*x])])]))*Sqrt[-(a*Cos[c + d*x])]*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - 2*
b^2 + 4*b*(b + a*Cos[c + d*x]) - 2*(b + a*Cos[c + d*x])^2)))*Sec[c + d*x]^(3/2))/(a*(a - b)^2*(a + b)^2*d*(a +
b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*((a^4 + a^2*b^2 + 4*b^4)/(2*a^2*(-a^2 + b^2)^2) - (2*b^5)/(a
^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + ((-a^2 - b^2 + 2*a*b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(-a^2 + b^2)^2)
)*Sec[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^(3/2))

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fricas [B]  time = 76.76, size = 8098, normalized size = 34.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b
- 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*l
og(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*
cos(d*x + c) + b)/cos(d*x + c))) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*
b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2
*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-(
(8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a
*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4
*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^
4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6
*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2
+ b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*
(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x
+ c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*
cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^
3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b -
3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3
*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3
*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2
*a*cos(d*x + c) + b)) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b
^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d
*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*
a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x +
c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^6*b - 5*
a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x
+ c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*
b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*(
(2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*
b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a
^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c
) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b
^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 +
3*a^4*b^5 - a^2*b^7)*d), -1/16*(2*(4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*
b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2
*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a + b)*arcta
n(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + 8*(a^6*
b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 +
3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d
*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) +
b)/cos(d*x + c))) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 +
17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x +
c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b
+ b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) +
b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b
^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*
cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)
^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*co
s(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a
^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a
^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x
+ c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4
*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^
3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(
-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) +
b)) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 -
7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7
- 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x
+ c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x +
c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*
cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*
sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b -
3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (
a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), 1/16*(2*(4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5
- (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 +
17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*
sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c
) + b)) - 8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6
*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)
*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((
a*cos(d*x + c) + b)/cos(d*x + c))) + (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^
6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a
^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(
-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt(
(a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) +
8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^
3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^
6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^
4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a
^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*co
s(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x
+ c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4
- 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 -
9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*co
s(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)
*cos(d*x + c) + b)) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2
- 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x
+ c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*
b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c)
+ b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3
*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6
)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x +
c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*
cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/8*((4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b
^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2
- 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*
cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a -
b)*cos(d*x + c) + b)) - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*
b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(
d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-
a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 4*(a^6*b - 3*a^4*b
^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 -
b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 -
8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x
+ c))) - 4*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5
*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4
- a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 +
3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/8*(8*(a^6*b - 3*a^4*b^
3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 -
b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*c
os(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^
3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b
^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^
4)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*
a - b)*cos(d*x + c) + b)) - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a
^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*c
os(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqr
t(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*((a^7 + 3*a^3
*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6
)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x +
c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*
cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d)]

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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(cot(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),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: (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: (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: (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: (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: (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: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Warning, integration of abs or sign assumes constant sign by intervals (correct if the argum
ent is real):Check [abs(cos(d*t_nostep+c))]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)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Una
ble 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)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)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Discontinuities at zeroes of cos(d*t_nostep
+c) were not checkedWarning, integration of abs or sign assumes constant sign by intervals (correct if the arg
ument is real):Check [abs(t_nostep^2-1)]Evaluation time: 1.2Unable to divide, perhaps due to rounding error%%%
{%%%{32,[2,6]%%%}+%%%{-32,[1,7]%%%},[6,1]%%%}+%%%{%%{[%%%{64,[2,6]%%%}+%%%{64,[1,7]%%%},0]:[1,0,%%%{-1,[1,0]%%
%}+%%%{1,[0,1]%%%}]%%},[5,1]%%%}+%%%{%%%{-32,[3,6]%%%}+%%%{64,[2,7]%%%}+%%%{96,[1,8]%%%},[4,1]%%%}+%%%{%%{[%%%
{-128,[3,6]%%%}+%%%{-256,[2,7]%%%}+%%%{-128,[1,8]%%%},0]:[1,0,%%%{-1,[1,0]%%%}+%%%{1,[0,1]%%%}]%%},[3,1]%%%}+%
%%{%%%{-32,[4,6]%%%}+%%%{-160,[3,7]%%%}+%%%{-224,[2,8]%%%}+%%%{-96,[1,9]%%%},[2,1]%%%}+%%%{%%{[%%%{64,[4,6]%%%
}+%%%{192,[3,7]%%%}+%%%{192,[2,8]%%%}+%%%{64,[1,9]%%%},0]:[1,0,%%%{-1,[1,0]%%%}+%%%{1,[0,1]%%%}]%%},[1,1]%%%}+
%%%{%%%{32,[5,6]%%%}+%%%{128,[4,7]%%%}+%%%{192,[3,8]%%%}+%%%{128,[2,9]%%%}+%%%{32,[1,10]%%%},[0,1]%%%} / %%%{%
%%{1,[2,0]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,2]%%%},[6,0]%%%}+%%%{%%{[%%%{2,[2,0]%%%}+%%%{-2,[0,2]%%%},0]:[1,0,%%%
{-1,[1,0]%%%}+%%%{1,[0,1]%%%}]%%},[5,0]%%%}+%%%{%%%{-1,[3,0]%%%}+%%%{3,[2,1]%%%}+%%%{1,[1,2]%%%}+%%%{-3,[0,3]%
%%},[4,0]%%%}+%%%{%%{[%%%{-4,[3,0]%%%}+%%%{-4,[2,1]%%%}+%%%{4,[1,2]%%%}+%%%{4,[0,3]%%%},0]:[1,0,%%%{-1,[1,0]%%
%}+%%%{1,[0,1]%%%}]%%},[3,0]%%%}+%%%{%%%{-1,[4,0]%%%}+%%%{-4,[3,1]%%%}+%%%{-2,[2,2]%%%}+%%%{4,[1,3]%%%}+%%%{3,
[0,4]%%%},[2,0]%%%}+%%%{%%{[%%%{2,[4,0]%%%}+%%%{4,[3,1]%%%}+%%%{-4,[1,3]%%%}+%%%{-2,[0,4]%%%},0]:[1,0,%%%{-1,[
1,0]%%%}+%%%{1,[0,1]%%%}]%%},[1,0]%%%}+%%%{%%%{1,[5,0]%%%}+%%%{3,[4,1]%%%}+%%%{2,[3,2]%%%}+%%%{-2,[2,3]%%%}+%%
%{-3,[1,4]%%%}+%%%{-1,[0,5]%%%},[0,0]%%%} Error: Bad Argument Value

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maple [B]  time = 1.96, size = 10977, normalized size = 46.51 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),x)

[Out]

result too large to display

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3/(a + b/cos(c + d*x))^(3/2),x)

[Out]

int(cot(c + d*x)^3/(a + b/cos(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Integral(cot(c + d*x)**3/(a + b*sec(c + d*x))**(3/2), x)

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