Integrand size = 23, antiderivative size = 251 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {107 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{64 \sqrt {2} \sqrt {a} d}+\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{64 a d}+\frac {43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac {15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d} \]
43/96*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^2/d-15/32*cos(d*x+c)*cot(d*x+c )^3*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(3/2)/a^2/d-1/16*cos(d*x+c)^2*co t(d*x+c)^3*sec(1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^(3/2)/a^2/d+2*arctan(a^(1 /2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d/a^(1/2)-107/128*arctan(1/2*a^(1/2 )*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))/d*2^(1/2)/a^(1/2)+21/64*cot(d *x+c)*(a+a*sec(d*x+c))^(1/2)/a/d
Time = 1.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {-321 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+\frac {-\left (\frac {1}{1+\cos (c+d x)}\right )^{3/2} (-110+19 \cos (c+d x)+142 \cos (2 (c+d x))+205 \cos (3 (c+d x))) \csc ^3\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right )+6144 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}}{8 \sqrt {2}}}{192 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {a (1+\sec (c+d x))}} \]
(-321*ArcSin[Tan[(c + d*x)/2]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 + Sec[c + d*x])^ (-1)]*Sqrt[1 + Sec[c + d*x]] + (-(((1 + Cos[c + d*x])^(-1))^(3/2)*(-110 + 19*Cos[c + d*x] + 142*Cos[2*(c + d*x)] + 205*Cos[3*(c + d*x)])*Csc[(c + d* x)/2]^3*Sec[(c + d*x)/2]) + 6144*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Se c[c + d*x]])/(8*Sqrt[2]))/(192*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a*(1 + Sec[ c + d*x])])
Time = 0.40 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4375, 374, 27, 441, 25, 27, 445, 27, 445, 27, 397, 216}
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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{\sqrt {a \sec (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle -\frac {2 \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle -\frac {2 \left (\frac {\int \frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (1-\frac {7 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (1-\frac {7 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {\int -\frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {75 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+43\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {\int \frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {75 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+43\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {1}{4} \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {75 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+43\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{6} \int -\frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (21-\frac {43 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (21-\frac {43 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+149\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+149}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (128 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-107 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {107 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {128 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
(-2*((Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(8*(2 + (a*Tan[c + d*x]^2 )/(a + a*Sec[c + d*x]))^2) + (((-43*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3 /2))/6 - (a*(-1/2*(a*((-128*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (107*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]))) + (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2)/4 + (15*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(4*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c + d*x]))))/8))/(a^2*d)
3.2.81.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d) Subst[Int[x^m*((2 + a*x^2 )^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] ]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I ntegerQ[n - 1/2]
Time = 2.10 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (321 \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+321 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-768 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-768 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+410 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}+142 \cot \left (d x +c \right )^{3}-298 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )-126 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{384 d a \left (\cos \left (d x +c \right )+1\right )}\) | \(372\) |
-1/384/d/a*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(321*2^(1/2)*ln(csc(d*x +c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2) )*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+321*ln(csc(d*x+c)-cot(d*x+ c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*2^(1/2)*(- cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-768*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ar ctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d* x+c)-768*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c) +1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+410*cos(d*x+c)*cot(d*x+c)^3+142*co t(d*x+c)^3-298*cot(d*x+c)^2*csc(d*x+c)-126*cot(d*x+c)*csc(d*x+c)^2)
Time = 0.36 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.65 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\left [-\frac {321 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 384 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, {\left (205 \, \cos \left (d x + c\right )^{4} + 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} - 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{768 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )}, \frac {321 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 384 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (205 \, \cos \left (d x + c\right )^{4} + 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} - 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{384 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )}\right ] \]
[-1/768*(321*sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)* sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))* cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(co s(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 384*(cos(d*x + c)^3 + c os(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2 *cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(205*cos(d*x + c)^4 + 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 - 63*c os(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*sin(d*x + c)), 1/384*(321* sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*arcta n(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*si n(d*x + c)))*sin(d*x + c) + 384*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c) )*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin (d*x + c) + 2*(205*cos(d*x + c)^4 + 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 - 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*sin(d*x + c))]
\[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
\[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \]
Time = 1.21 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.53 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {2} {\left (6 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (\frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a} - \frac {21}{a}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {384 \, \sqrt {2} \sqrt {-a} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} + \frac {321 \, \log \left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2}\right )}{\sqrt {-a}} - \frac {64 \, {\left (9 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} \sqrt {-a} - 15 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} \sqrt {-a} a + 8 \, \sqrt {-a} a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3}}\right )}}{768 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
1/768*sqrt(2)*(6*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*tan(1/2*d*x + 1/2* c)^2/a - 21/a)*tan(1/2*d*x + 1/2*c) - 384*sqrt(2)*sqrt(-a)*log(abs(2*(sqrt (-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 4*sqr t(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2* d*x + 1/2*c)^2 + a))^2 + 4*sqrt(2)*abs(a) - 6*a))/abs(a) + 321*log((sqrt(- a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/sqrt(-a) - 64*(9*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*sqrt(-a) - 15*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*sqrt(-a)*a + 8*sqrt(-a)*a^2)/((sqrt(-a)*tan(1/2*d*x + 1 /2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^3)/(d*sgn(cos(d*x + c) ))
Timed out. \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]