Integrand size = 23, antiderivative size = 215 \[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {a \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 \sqrt {a-b} d}+\frac {a \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 \sqrt {a+b} d}-\frac {\cot ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d} \]
-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
Time = 2.45 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.99 \[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\frac {-\frac {b \arctan \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {-a+b}}\right )}{\sqrt {-a+b}}-8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )+\frac {4 \sqrt {-(a-b)^2} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {-a+b}}+\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-2 \cot ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d} \]
(-((b*ArcTan[Sqrt[a + b*Sec[c + d*x]]/Sqrt[-a + b]])/Sqrt[-a + b]) - 8*Sqr t[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]] + (4*Sqrt[-(a - b)^2]*ArcTa nh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/Sqrt[-a + b] + (4*a*ArcTanh[Sqrt [a + b*Sec[c + d*x]]/Sqrt[a + b]])/Sqrt[a + b] + (3*b*ArcTanh[Sqrt[a + b*S ec[c + d*x]]/Sqrt[a + b]])/Sqrt[a + b] - 2*Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]])/(4*d)
Time = 0.55 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 25, 4373, 561, 25, 1652, 25, 1484, 1492, 27, 1406, 220, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\cot \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sqrt {a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle \frac {b^4 \int \frac {\cos (c+d x) \sqrt {a+b \sec (c+d x)}}{b \left (b^2-b^2 \sec ^2(c+d x)\right )^2}d(b \sec (c+d x))}{d}\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {2 b^4 \int -\frac {b^2 \sec ^2(c+d x)}{\left (a-b^2 \sec ^2(c+d x)\right ) \left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )^2}d\sqrt {a+b \sec (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b^4 \int \frac {b^2 \sec ^2(c+d x)}{\left (a-b^2 \sec ^2(c+d x)\right ) \left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )^2}d\sqrt {a+b \sec (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1652 |
| \(\displaystyle \frac {2 b^4 \left (\frac {\int -\frac {a^2-b^2 \sec ^2(c+d x) a-b^2}{\left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )^2}d\sqrt {a+b \sec (c+d x)}}{b^2}+\frac {a \int \frac {1}{\left (a-b^2 \sec ^2(c+d x)\right ) \left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )}d\sqrt {a+b \sec (c+d x)}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \frac {1}{\left (a-b^2 \sec ^2(c+d x)\right ) \left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )}d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\int \frac {a^2-b^2 \sec ^2(c+d x) a-b^2}{\left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )^2}d\sqrt {a+b \sec (c+d x)}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \left (\frac {1}{2 b^2 \left (-b^2 \sec ^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \sec ^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}\right )d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\int \frac {a^2-b^2 \sec ^2(c+d x) a-b^2}{\left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )^2}d\sqrt {a+b \sec (c+d x)}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \left (\frac {1}{2 b^2 \left (-b^2 \sec ^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \sec ^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}\right )d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\frac {\sqrt {a+b \sec (c+d x)}}{4 \left (a^2-2 a b^2 \sec ^2(c+d x)+b^4 \sec ^4(c+d x)-b^2\right )}-\frac {\int -\frac {6 b^2 \left (a^2-b^2\right )}{b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2}d\sqrt {a+b \sec (c+d x)}}{8 b^2 \left (a^2-b^2\right )}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \left (\frac {1}{2 b^2 \left (-b^2 \sec ^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \sec ^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}\right )d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\frac {3}{4} \int \frac {1}{b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2}d\sqrt {a+b \sec (c+d x)}+\frac {\sqrt {a+b \sec (c+d x)}}{4 \left (a^2-2 a b^2 \sec ^2(c+d x)+b^4 \sec ^4(c+d x)-b^2\right )}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \left (\frac {1}{2 b^2 \left (-b^2 \sec ^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \sec ^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}\right )d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\frac {3}{4} \left (\frac {\int \frac {1}{b^2 \sec ^2(c+d x)-a-b}d\sqrt {a+b \sec (c+d x)}}{2 b}-\frac {\int \frac {1}{b^2 \sec ^2(c+d x)-a+b}d\sqrt {a+b \sec (c+d x)}}{2 b}\right )+\frac {\sqrt {a+b \sec (c+d x)}}{4 \left (a^2-2 a b^2 \sec ^2(c+d x)+b^4 \sec ^4(c+d x)-b^2\right )}}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \int \left (\frac {1}{2 b^2 \left (-b^2 \sec ^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \sec ^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}\right )d\sqrt {a+b \sec (c+d x)}}{b^2}-\frac {\frac {\sqrt {a+b \sec (c+d x)}}{4 \left (a^2-2 a b^2 \sec ^2(c+d x)+b^4 \sec ^4(c+d x)-b^2\right )}+\frac {3}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{2 b \sqrt {a-b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{2 b \sqrt {a+b}}\right )}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^4 \left (\frac {a \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{2 b^2 \sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{2 b^2 \sqrt {a+b}}\right )}{b^2}-\frac {\frac {\sqrt {a+b \sec (c+d x)}}{4 \left (a^2-2 a b^2 \sec ^2(c+d x)+b^4 \sec ^4(c+d x)-b^2\right )}+\frac {3}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{2 b \sqrt {a-b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{2 b \sqrt {a+b}}\right )}{b^2}\right )}{d}\) |
(2*b^4*((a*(-(ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]]/(Sqrt[a]*b^2)) + A rcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(2*Sqrt[a - b]*b^2) + ArcTanh [Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(2*b^2*Sqrt[a + b])))/b^2 - ((3*(Ar cTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(2*Sqrt[a - b]*b) - ArcTanh[Sq rt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(2*b*Sqrt[a + b])))/4 + Sqrt[a + b*Sec [c + d*x]]/(4*(a^2 - b^2 - 2*a*b^2*Sec[c + d*x]^2 + b^4*Sec[c + d*x]^4)))/ b^2))/d
3.4.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[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]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{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]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> 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] + Simp[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]
Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[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] - Simp[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]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-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]
Leaf count of result is larger than twice the leaf count of optimal. \(2071\) vs. \(2(177)=354\).
Time = 2.20 (sec) , antiderivative size = 2072, normalized size of antiderivative = 9.64
1/8/d/(a-b)^(3/2)/(a+b)*(-((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^ 4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^(3/2)* a*(1-cos(d*x+c))^4*csc(d*x+c)^4+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d* x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^ (3/2)*b*(1-cos(d*x+c))^4*csc(d*x+c)^4+8*a^(3/2)*ln(2*(-2*a*(1-cos(d*x+c))^ 2*csc(d*x+c)^2+b*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1-cos(d*x+c))^4 *a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d *x+c)^2+a+b)^(1/2)+2*a+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*(1-cos(d*x+c) )^2*(a-b)^(3/2)*csc(d*x+c)^2+8*a^(1/2)*ln(2*(-2*a*(1-cos(d*x+c))^2*csc(d*x +c)^2+b*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1-cos(d*x+c))^4*a*csc(d* x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a +b)^(1/2)+2*a+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*(1-cos(d*x+c))^2*(a-b) ^(3/2)*b*csc(d*x+c)^2-4*(a+b)^(1/2)*ln(2*(-a*(1-cos(d*x+c))^2*csc(d*x+c)^2 +(a+b)^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c )^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+a+b)/(1-cos(d*x+c))^2*sin (d*x+c)^2)*(1-cos(d*x+c))^2*(a-b)^(3/2)*a*csc(d*x+c)^2-3*(a+b)^(1/2)*ln(2* (-a*(1-cos(d*x+c))^2*csc(d*x+c)^2+(a+b)^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+ c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b )^(1/2)+a+b)/(1-cos(d*x+c))^2*sin(d*x+c)^2)*(1-cos(d*x+c))^2*(a-b)^(3/2)*b *csc(d*x+c)^2+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d...
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (177) = 354\).
Time = 1.81 (sec) , antiderivative size = 3523, normalized size of antiderivative = 16.39 \[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
[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)*c os(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*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)*sq rt((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...
\[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int \sqrt {a + b \sec {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx \]
\[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (177) = 354\).
Time = 0.84 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.39 \[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\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} \]
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
Timed out. \[ \int \cot ^3(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]