Integrand size = 23, antiderivative size = 144 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {3 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d} \]
2*a^(3/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d-1/3*cot(d*x+ c)^3*(a+a*sec(d*x+c))^(3/2)/d-1/4*a^(3/2)*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)+3/2*a*cot(d*x+c)*(a+a*sec(d*x+c))^ (1/2)/d
Time = 6.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-4 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}}{8 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x)}+\frac {\cos (c+d x) \sec ^3\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{3/2} \left (\frac {13}{24} \csc \left (\frac {1}{2} (c+d x)\right )-\frac {1}{24} \csc ^3\left (\frac {1}{2} (c+d x)\right )-\frac {11}{12} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
-1/8*((ArcSin[Tan[(c + d*x)/2]] - 4*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[C os[c + d*x]/(1 + Cos[c + d*x])]])*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Se c[(c + d*x)/2]^4*Sqrt[1 + Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))/(d*S qrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*Sec[(c + d*x)/ 2]^3*(a*(1 + Sec[c + d*x]))^(3/2)*((13*Csc[(c + d*x)/2])/24 - Csc[(c + d*x )/2]^3/24 - (11*Sin[(c + d*x)/2])/12))/d
Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4375, 382, 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.
| \(\displaystyle \int \cot ^4(c+d x) (a \sec (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle -\frac {2 \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\) |
| \(\Big \downarrow \) 382 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \int -\frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+3\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+3\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )}{d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {3}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+7\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {3}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+7}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {3}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (4 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \left (\frac {1}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {3}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {4 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )\right )}{d}\) |
(-2*((Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/6 - (a*(-1/2*(a*((-4*ArcT an[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + ArcTan[(Sqr t[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a]))) + (3*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2))/d
3.2.57.3.1 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[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && LtQ[m, -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[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 = 1.74 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(-\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-24 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+22 \cot \left (d x +c \right )^{3}+4 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )-18 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{12 d}\) | \(197\) |
-1/12/d*a*(a*(1+sec(d*x+c)))^(1/2)*(3*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)/(co s(d*x+c)+1))^(1/2)-24*(-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))+22*cot(d*x+c)^3+4*cot (d*x+c)^2*csc(d*x+c)-18*cot(d*x+c)*csc(d*x+c)^2)
Time = 0.40 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.69 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\left [\frac {3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\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 ) + 12 \, {\left (a \cos \left (d x + c\right ) - a\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 (11 \, a \cos \left (d x + c\right )^{2} - 9 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{24 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}, \frac {12 \, {\left (a \cos \left (d x + c\right ) - a\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 ) + 3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\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 ) + 2 \, {\left (11 \, a \cos \left (d x + c\right )^{2} - 9 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{12 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}\right ] \]
[1/24*(3*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*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)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 12*(a*cos(d*x + c) - a)*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*(11*a*cos(d*x + c)^2 - 9*a*cos(d*x + c))*sqrt((a*cos(d* x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c)), 1/12*(12*( a*cos(d*x + c) - a)*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) + 3*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*sqrt(a)*arctan (sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin (d*x + c)))*sin(d*x + c) + 2*(11*a*cos(d*x + c)^2 - 9*a*cos(d*x + c))*sqrt ((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c))]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]