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

Optimal. Leaf size=215 \[ -\frac {\cot ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 d \sqrt {a-b}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 d \sqrt {a+b}} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3885, 898, 1315, 1178, 12, 1093, 206, 1170, 207} \[ -\frac {\cot ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 d \sqrt {a-b}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d + (a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(
Sqrt[a - b]*d) - (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) + (a*ArcTanh[Sqrt[a + b
*Sec[c + d*x]]/Sqrt[a + b]])/(Sqrt[a + b]*d) + (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a +
b]*d) - (Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]])/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

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

Rule 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])

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 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1170

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

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1315

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[
f^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x], x] - Dist[(d*e*f^2)/
(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 0]

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 \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx &=\frac {b^4 \operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{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 {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^2\right ) \operatorname {Subst}\left (\int \frac {-a^2+b^2+a x^2}{\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 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}\\ &=-\frac {b^2 \sqrt {a+b \sec (c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \sec (c+d x))+(a+b \sec (c+d x))^2\right )}+\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{b^2 \left (a-x^2\right )}+\frac {1}{2 b^2 \left (a+b-x^2\right )}-\frac {1}{2 b^2 \left (-a+b+x^2\right )}\right ) \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {6 b^2 \left (a^2-b^2\right )}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac {b^2 \sqrt {a+b \sec (c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \sec (c+d x))+(a+b \sec (c+d x))^2\right )}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{2 d}\\ &=-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {b^2 \sqrt {a+b \sec (c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \sec (c+d x))+(a+b \sec (c+d x))^2\right )}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{4 d}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{4 d}\\ &=-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 \sqrt {a-b} d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 \sqrt {a+b} d}-\frac {b^2 \sqrt {a+b \sec (c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \sec (c+d x))+(a+b \sec (c+d x))^2\right )}\\ \end {align*}

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Mathematica [B]  time = 18.95, size = 937, normalized size = 4.36 \[ \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {1}{2}-\frac {1}{2} \csc ^2(c+d x)\right )}{d}+\frac {\left (\frac {2 \left (4 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b+a \cos (c+d x)}}{\sqrt {-a \cos (c+d x)}}\right )-\sqrt {a} \left (\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a-b} \sqrt {-a \cos (c+d x)}}\right )+\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a+b} \sqrt {-a \cos (c+d x)}}\right )\right )\right ) \sqrt {-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt {\sec (c+d x)} a^2}{\sqrt {a-b} \sqrt {a+b} \left (a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right )}+\frac {3 b \left (-\sqrt {-a^2} \sqrt {a+b} \log \left (\sqrt {b+a \cos (c+d x)}-\sqrt {b-a}\right )+\sqrt {-a^2} \sqrt {a+b} \log \left (\sqrt {b-a}+\sqrt {b+a \cos (c+d x)}\right )-a \sqrt {b-a} \log \left (\sqrt {b+a \cos (c+d x)}-\sqrt {a+b}\right )+a \sqrt {b-a} \log \left (\sqrt {a+b}+\sqrt {b+a \cos (c+d x)}\right )+\sqrt {-a^2} \sqrt {a+b} \log \left (b+\sqrt {a} \sqrt {-a \cos (c+d x)}-\sqrt {b-a} \sqrt {b+a \cos (c+d x)}\right )-\sqrt {-a^2} \sqrt {a+b} \log \left (b+\sqrt {a} \sqrt {-a \cos (c+d x)}+\sqrt {b-a} \sqrt {b+a \cos (c+d x)}\right )+a \sqrt {b-a} \log \left (b+\sqrt {-a} \sqrt {-a \cos (c+d x)}-\sqrt {a+b} \sqrt {b+a \cos (c+d x)}\right )-a \sqrt {b-a} \log \left (b+\sqrt {-a} \sqrt {-a \cos (c+d x)}+\sqrt {a+b} \sqrt {b+a \cos (c+d x)}\right )\right ) a}{2 (-a)^{3/2} \sqrt {b-a} \sqrt {a+b} \sqrt {-a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {2 \left (\sqrt {a-b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a-b} \sqrt {-a \cos (c+d x)}}\right )+(a-b) \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b+a \cos (c+d x)}}{\sqrt {a+b} \sqrt {-a \cos (c+d x)}}\right )\right ) \sqrt {-a \cos (c+d x)} \sqrt {\sec (c+d x)} \sqrt {a}}{(a-b) (a+b)}\right ) \sqrt {a+b \sec (c+d x)}}{4 d \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1/2 - Csc[c + d*x]^2/2)*Sqrt[a + b*Sec[c + d*x]])/d + (((3*a*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]]] - S
qrt[-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*Sqr
t[-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]*Lo
g[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]*Sq
rt[a + b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a]*(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*Co
s[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/((a - b)*(a +
b)) + (2*a^2*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(S
qrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcT
an[(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[a + b*Sec[c + d*x]])/(4*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])

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fricas [B]  time = 3.14, size = 3523, normalized size = 16.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 8*((a^2 - b^2)*cos(d*x + c)^2 -
a^2 + b^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^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3
*b^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^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^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)))/((a^2 - b^2)
*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2
- 2*((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*c
os(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^
2 + b^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^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b
^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*c
os(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) +
b)/cos(d*x + c))*cos(d*x + c)^2 + 2*((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^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^2 - b^2)*cos(d*x + c)^2 - a^2 + b^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^2 - a*b - 3*b^2)*co
s(d*x + c)^2 - 4*a^2 + a*b + 3*b^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)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/8*(4*(a^2
- b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2
- a*b + 3*b^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^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^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^2 - b^
2)*cos(d*x + c)^2 - a^2 + b^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))))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2
- b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d
*x + c)^2 - a^2 + b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*co
s(d*x + c) + b)) - ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^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^2 - a*b
- 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^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)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d),
1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d*x + c)^2 -
a^2 + b^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^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^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^2 + a*b - 3*b^2)*cos(d*x
+ c)^2 - 4*a^2 - a*b + 3*b^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)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b
^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^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^2
+ a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^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^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2
+ a*b + 3*b^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)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/8*(4*(a^2 - b^2)*sqrt((a*cos(
d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^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^2 + a*b - 3*b^2)*cos
(d*x + c)^2 - 4*a^2 - a*b + 3*b^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^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqr
t(-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)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d)]

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giac [B]  time = 1.89, size = 514, normalized size = 2.39 \[ \frac {{\left (\frac {16 \, a \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2 \, {\left (4 \, a + 3 \, b\right )} \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b}} + \frac {{\left (4 \, a - 3 \, b\right )} \log \left ({\left | {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} {\left (a - b\right )} - \sqrt {a - b} a \right |}\right )}{\sqrt {a - b}} + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} - \frac {2 \, {\left ({\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} a - {\left (a + b\right )} \sqrt {a - b}\right )}}{{\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{2} - a - b}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{8 \, d} \]

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]

1/8*(16*a*arctan(-1/2*(sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/
2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b) + sqrt(a - b))/sqrt(-a))/sqrt(-a) - 2*(4*a + 3*b)*arctan(-(sqrt(a
- b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/
2*c)^2 + a + b))/sqrt(-a - b))/sqrt(-a - b) + (4*a - 3*b)*log(abs((sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a
*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b))*(a - b) - sqrt(a - b
)*a))/sqrt(a - b) + sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a
+ b) - 2*((sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a
*tan(1/2*d*x + 1/2*c)^2 + a + b))*a - (a + b)*sqrt(a - b))/((sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1
/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b))^2 - a - b))*sgn(cos(d*x +
c))/d

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maple [B]  time = 1.48, size = 2844, normalized size = 13.23 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/d*(-1+cos(d*x+c))*(-4*(a-b)^(3/2)*(a+b)^(1/2)*cos(d*x+c)*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x
+c))^2)^(1/2)*a-16*(a-b)^(3/2)*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(3/2)+4*(a+b)^(1/2)*
cos(d*x+c)^2*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))
^2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/si
n(d*x+c)^2/(a-b)^(1/2))*a^3+3*(a+b)^(1/2)*cos(d*x+c)^2*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*(
(b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)
/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2/(a-b)^(1/2))*b^3+7*(a-b)^(3/2)*4^(1/2)*ln(-2*(2*cos(d*x+c
)*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^(1/2)*(
(b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+b)/(-1+cos(d*x+c)))*a*b+3*(a+b)^(1/2)*4^(1/2)*ln(-(-1+cos(
d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*
x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2/(a-b)^(1/2))*a^2*b-16*
(a-b)^(3/2)*(a+b)^(1/2)*cos(d*x+c)^2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(3/2)-32*(a-b)^(3/2)*(a+b)
^(1/2)*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(3/2)+4*(a-b)^(3/2)*4^(1/2)*ln(-2*(2*cos(d*x+
c)*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^(1/2)*
((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+b)/(-1+cos(d*x+c)))*a^2+3*(a-b)^(3/2)*4^(1/2)*ln(-2*(2*co
s(d*x+c)*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^
(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+b)/(-1+cos(d*x+c)))*b^2-4*(a+b)^(1/2)*4^(1/2)*ln(-(
-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)-2*a*cos(d*x+c)+b
*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2/(a-b)^(1/2))*a^
3-3*(a+b)^(1/2)*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+
c))^2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)
/sin(d*x+c)^2/(a-b)^(1/2))*b^3+4*(a-b)^(3/2)*(a+b)^(1/2)*cos(d*x+c)*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+co
s(d*x+c))^2)^(1/2)*b+8*(a-b)^(3/2)*(a+b)^(1/2)*a^(1/2)*cos(d*x+c)^2*4^(1/2)*ln(4*a^(1/2)*cos(d*x+c)*((b+a*cos(
d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a*
cos(d*x+c)+2*b)*b+4*(a-b)^(3/2)*(a+b)^(1/2)*cos(d*x+c)^2*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2
)^(1/2)*a-4*(a-b)^(3/2)*(a+b)^(1/2)*cos(d*x+c)^2*4^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*
b+8*(a-b)^(3/2)*(a+b)^(1/2)*a^(3/2)*cos(d*x+c)^2*4^(1/2)*ln(4*a^(1/2)*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/
(1+cos(d*x+c))^2)^(1/2)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a*cos(d*x+c)+2*b)+4*(
a+b)^(1/2)*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2
)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin(
d*x+c)^2/(a-b)^(1/2))*a*b^2-8*(a-b)^(3/2)*(a+b)^(1/2)*a^(3/2)*4^(1/2)*ln(4*a^(1/2)*cos(d*x+c)*((b+a*cos(d*x+c)
)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a*cos(d*
x+c)+2*b)-4*(a-b)^(3/2)*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*cos(d*x+c)*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+c
os(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(
1/2)+b)/(-1+cos(d*x+c)))*a^2-3*(a-b)^(3/2)*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*cos(d*x+c)*(a+b)^(1/2)*((b+a*cos(d*x+
c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/
(1+cos(d*x+c))^2)^(1/2)+b)/(-1+cos(d*x+c)))*b^2-7*(a-b)^(3/2)*cos(d*x+c)^2*4^(1/2)*ln(-2*(2*cos(d*x+c)*(a+b)^(
1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+2*a*cos(d*x+c)+b*cos(d*x+c)+2*(a+b)^(1/2)*((b+a*cos(
d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+b)/(-1+cos(d*x+c)))*a*b-8*(a-b)^(3/2)*(a+b)^(1/2)*a^(1/2)*4^(1/2)*l
n(4*a^(1/2)*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*
x+c)/(1+cos(d*x+c))^2)^(1/2)+4*a*cos(d*x+c)+2*b)*b-3*(a+b)^(1/2)*cos(d*x+c)^2*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*c
os(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a
*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2/(a-b)^(1/2))*a^2*b-4*(a+b)^(1/2)*c
os(d*x+c)^2*4^(1/2)*ln(-(-1+cos(d*x+c))*(2*cos(d*x+c)*(a-b)^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^
2)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(a-b)^(1/2)-b)/sin
(d*x+c)^2/(a-b)^(1/2))*a*b^2)*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)/sin(d*x+c)^4/((b+a*cos(d*x+c))*co
s(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(a+b)^(3/2)/(a-b)^(3/2)

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3*(a + b/cos(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^3*(a + b/cos(c + d*x))^(1/2), x)

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

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

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