3.330 \(\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=260 \[ \frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right )}{4 d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right )}{d \sqrt{a-b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )}{d \sqrt{a+b}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )}{4 d (a+b)^{3/2}} \]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sq
rt[a - b]*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + (b*ArcTanh[Sqrt[a + b*S
ec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)
+ Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)*d*(1 + Sec[
c + d*x]))

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Rubi [A]  time = 0.263382, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3885, 898, 1238, 206, 199, 207} \[ \frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right )}{4 d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right )}{d \sqrt{a-b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )}{d \sqrt{a+b}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )}{4 d (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sq
rt[a - b]*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + (b*ArcTanh[Sqrt[a + b*S
ec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)
+ Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)*d*(1 + Sec[
c + d*x]))

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]

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 1238

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

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 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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [B]  time = 6.90347, size = 1022, normalized size = 3.93 \[ \frac{(b+a \cos (c+d x)) \left (\frac{(a-b \cos (c+d x)) \csc ^2(c+d x)}{2 \left (b^2-a^2\right )}+\frac{a}{2 \left (a^2-b^2\right )}\right ) \sec (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \left (-\frac{b \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 ) a^2}{2 (-a)^{3/2} \sqrt{b-a} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (2 a^2-2 b^2\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)} a}{\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{\left (2 a^2-3 b^2\right ) \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 \cos (c+d x)} \sqrt{\sec (c+d x)}}{(a-b) (a+b) \sqrt{a}}\right ) \sqrt{\sec (c+d x)}}{4 (a-b) (a+b) d \sqrt{a+b \sec (c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[b + a*Cos[c + d*x]]*(-(a^2*b*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + Sqrt[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]]] - 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]]] + a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sq
rt[-(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]]]))/(2*(-a)^(3/2)*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[-(a*Cos[c + d*
x])]*Sqrt[Sec[c + d*x]]) - ((2*a^2 - 3*b^2)*(Sqrt[a - b]*(a + b)*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sq
rt[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^
2 - 2*b^2)*(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)))*Sqrt[Sec[c + d*x]])/(4*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*(a/(2*(
a^2 - b^2)) + ((a - b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(-a^2 + b^2)))*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]
])

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Maple [B]  time = 0.409, size = 4203, normalized size = 16.2 \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)^3/(a+b*sec(d*x+c))^(1/2),x)

[Out]

-1/16/d/a/(a-b)^(5/2)/(a+b)^(5/2)*(-1+cos(d*x+c))*(4*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)
+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*a*cos
(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^4-16*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/
2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2-4*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos
(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*
x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^5+8*cos(d*x+c)^2*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos
(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos
(d*x+c)+1)^2)^(1/2)+2*b)*(a-b)^(3/2)*a^(7/2)*(a+b)^(1/2)-4*cos(d*x+c)*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(co
s(d*x+c)+1)^2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^3-8*cos(d*x+c)^2*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(
d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)
^(1/2)+2*b)*(a-b)^(3/2)*a^(1/2)*(a+b)^(1/2)*b^3-12*cos(d*x+c)^2*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+
c)+1)^2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2*b+8*cos(d*x+c)*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^
2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2*b-4*cos(d*x+c)*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/
2)*(a-b)^(3/2)*(a+b)^(1/2)*a*b^2+8*cos(d*x+c)^2*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+
c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*(a-b
)^(3/2)*a^(5/2)*(a+b)^(1/2)*b-8*cos(d*x+c)^2*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+
1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*(a-b)^(
3/2)*a^(3/2)*(a+b)^(1/2)*b^2+4*cos(d*x+c)^2*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(
d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x
+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^5-32*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d
*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2+5*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos
(d*x+c)+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+
2*a*cos(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^3*b-4*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x
+c)/(cos(d*x+c)+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2
)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^2*b^2-5*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c
))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d
*x+c)+1)^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a*b^3+16*((b+a*cos(d*x+c))*cos(d
*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*(a+b)^(1/2)*a*b+4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x
+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d
*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^4*b+9*4^(1/2)*ln(-1/(a-b)
^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(
d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^
(1/2)*a^3*b^2-4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)
+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a
-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^2*b^3-5*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a
*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*co
s(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a*b^4+4*4^(1/2)*((b+a*cos(d*x+c))*co
s(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2*b+4*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x
+c)+1)^2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a*b^2+8*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+
c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*(a-b
)^(3/2)*a^(1/2)*(a+b)^(1/2)*b^3-8*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)
*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*(a-b)^(3/2)*a^(5/2
)*(a+b)^(1/2)*b-5*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a+b)^(1/
2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+b
)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^3*b+4*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+
1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*a*cos(
d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^2*b^2+5*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*((b+a*cos(d*x+c))*
cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+
c)+1)^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a*b^3+16*cos(d*x+c)^2*((b+a*cos(d*x
+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*(a+b)^(1/2)*a*b+8*4^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c)
)*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)
+1)^2)^(1/2)+2*b)*(a-b)^(3/2)*a^(3/2)*(a+b)^(1/2)*b^2-cos(d*x+c)^2*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(
2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((
b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^4*b-9*cos(d*x+c)
^2*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2
)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b
)/sin(d*x+c)^2)*(a+b)^(1/2)*a^3*b^2+cos(d*x+c)^2*4^(1/2)*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a
*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*co
s(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*(a+b)^(1/2)*a^2*b^3+5*cos(d*x+c)^2*4^(1/2)*ln(-1
/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*
a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*
(a+b)^(1/2)*a*b^4+32*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*(a+b)^(1/2)*a
*b+4*cos(d*x+c)^2*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^3-8*4
^(1/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*(
(b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*(a-b)^(3/2)*a^(7/2)*(a+b)^(1/2)-4*cos(d*x+c)^2*4^(1/2
)*ln(-2*(2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a+b)^(1/2)*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos
(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+b)/(-1+cos(d*x+c)))*(a-b)^(3/2)*a^4-16
*cos(d*x+c)^2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*(a+b)^(1/2)*a^2)*((b+a*cos(d*x+
c))/cos(d*x+c))^(1/2)*cos(d*x+c)/((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)/sin(d*x+c)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{3}}{\sqrt{b \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 145.021, size = 10157, normalized size = 39.07 \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)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*s
qrt(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))*sq
rt(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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos
(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(16*(a^4 - 2*a^2
*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6
*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b
- 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos
(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3
*b^2 + a*b^4)*d), -1/16*(2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*co
s(d*x + c)^2)*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^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos
(d*x + c)^2)*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*co
s(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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x
+ c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(1
6*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a
^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4
- 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b -
a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (
a^5 - 2*a^3*b^2 + a*b^4)*d), 1/16*(2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5
*a*b^3)*cos(d*x + c)^2)*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^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a
*b^3)*cos(d*x + c)^2)*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))/(co
s(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((
a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d),
-1/16*(16*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b
^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3
- (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - a^2*b^2)*cos(d*x + c)^2 -
(a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x +
c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/8*((4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b
^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2
+ 5*a*b^3)*cos(d*x + c)^2)*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^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt(
(a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d)
, -1/8*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3
- (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*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^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 -
(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*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^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b -
a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (
a^5 - 2*a^3*b^2 + a*b^4)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\left (c + d x \right )}}{\sqrt{a + b \sec{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3/(a+b*sec(d*x+c))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{3}}{\sqrt{b \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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