Integrand size = 23, antiderivative size = 259 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {107 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{64 \sqrt {2} \sqrt {a} d}+\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{64 a d}+\frac {43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{4 a^2 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {15 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{16 a^2 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:
2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(1/2)/d-107/128*arct an(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/a^(1/2)/ d+21/64*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a/d+43/96*cot(d*x+c)^3*(a+a*sec( d*x+c))^(3/2)/a^2/d-1/4*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^2/d/(2+tan(d *x+c)^2/(1+sec(d*x+c)))^2-15/16*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^2/d/ (2+tan(d*x+c)^2/(1+sec(d*x+c)))
Time = 1.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {-321 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+\frac {-\left (\frac {1}{1+\cos (c+d x)}\right )^{3/2} (-110+19 \cos (c+d x)+142 \cos (2 (c+d x))+205 \cos (3 (c+d x))) \csc ^3\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right )+6144 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}}{8 \sqrt {2}}}{192 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {a (1+\sec (c+d x))}} \] Input:
Integrate[Cot[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]
Output:
(-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 = 0.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.02, 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 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {107 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {128 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )-\frac {43}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {15 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
Input:
Int[Cot[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]
Output:
(-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)
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 = 0.87 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.62
| method | result | size |
| default | \(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (-1536 \cos \left (d x +c \right )^{3}-4608 \cos \left (d x +c \right )^{2}-4608 \cos \left (d x +c \right )-1536\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+\left (-1284 \cos \left (d x +c \right )^{3}-3852 \cos \left (d x +c \right )^{2}-3852 \cos \left (d x +c \right )-1284\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\sqrt {2}\, \left (485 \cos \left (d x +c \right )^{5}+3037 \cos \left (d x +c \right )^{4}+3542 \cos \left (d x +c \right )^{3}-858 \cos \left (d x +c \right )^{2}-3003 \cos \left (d x +c \right )-1155\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}+\left (-670 \cos \left (d x +c \right )^{5}+1256 \cos \left (d x +c \right )^{4}+396 \cos \left (d x +c \right )^{3}-1376 \cos \left (d x +c \right )^{2}-110 \cos \left (d x +c \right )+504\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{1536 d a \left (1+\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right )}\) | \(420\) |
Input:
int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/1536/d/a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^2+2*cos(d*x +c)+1)*((-1536*cos(d*x+c)^3-4608*cos(d*x+c)^2-4608*cos(d*x+c)-1536)*2^(1/2 )*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+cot(d* x+c))/(cot(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+csc(d*x+c)^2-1)^(1/2))+(-1284* cos(d*x+c)^3-3852*cos(d*x+c)^2-3852*cos(d*x+c)-1284)*(-2*cos(d*x+c)/(1+cos (d*x+c)))^(1/2)*ln((-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-cot(d*x+c)+csc(d*x +c))+2^(1/2)*(485*cos(d*x+c)^5+3037*cos(d*x+c)^4+3542*cos(d*x+c)^3-858*cos (d*x+c)^2-3003*cos(d*x+c)-1155)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-cos (d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)^2+(-670*cos(d*x+c)^5+1 256*cos(d*x+c)^4+396*cos(d*x+c)^3-1376*cos(d*x+c)^2-110*cos(d*x+c)+504)*co t(d*x+c)*csc(d*x+c)^2)
Time = 0.17 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.57 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[-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))]
\[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \] Input:
integrate(cot(d*x+c)**4/(a+a*sec(d*x+c))**(1/2),x)
Output:
Integral(cot(c + d*x)**4/sqrt(a*(sec(c + d*x) + 1)), x)
\[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(cot(d*x + c)^4/sqrt(a*sec(d*x + c) + a), x)
Time = 0.62 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {2} {\left (6 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (\frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a} - \frac {21}{a}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {384 \, \sqrt {2} \sqrt {-a} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} + \frac {321 \, \log \left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2}\right )}{\sqrt {-a}} - \frac {64 \, {\left (9 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} \sqrt {-a} - 15 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} \sqrt {-a} a + 8 \, \sqrt {-a} a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3}}\right )}}{768 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \] Input:
integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
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. \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(1/2),x)
Output:
int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(1/2), x)
\[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4}}{\sec \left (d x +c \right )+1}d x \right )}{a} \] Input:
int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x)**4)/(sec(c + d*x) + 1),x ))/a