Integrand size = 23, antiderivative size = 288 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {151 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {105 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 d}-\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{4 a^2 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {17 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{16 a^2 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:
-2*a^(1/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+151/256*a^( 1/2)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2) /d-105/128*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d-23/192*cot(d*x+c)^3*(a+a*se c(d*x+c))^(3/2)/a/d+87/160*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^2/d-1/4*c ot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^2/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))^2 -17/16*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^2/d/(2+tan(d*x+c)^2/(1+sec(d* x+c)))
Time = 3.64 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\sqrt {a (1+\sec (c+d x))} \left (-\frac {(5207+172 \cos (c+d x)-4572 \cos (2 (c+d x))-556 \cos (3 (c+d x))+2821 \cos (4 (c+d x))) \csc ^5(c+d x)}{4 \sqrt {\sec (c+d x)}}-7680 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}+2265 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{3840 d \sqrt {\sec (c+d x)}} \] Input:
Integrate[Cot[c + d*x]^6*Sqrt[a + a*Sec[c + d*x]],x]
Output:
(Sqrt[a*(1 + Sec[c + d*x])]*(-1/4*((5207 + 172*Cos[c + d*x] - 4572*Cos[2*( c + d*x)] - 556*Cos[3*(c + d*x)] + 2821*Cos[4*(c + d*x)])*Csc[c + d*x]^5)/ Sqrt[Sec[c + d*x]] - 7680*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^ (-1)]]*Sqrt[Sec[c + d*x]/(1 + Sec[c + d*x])^2]*Sqrt[1 + Sec[c + d*x]] + 22 65*ArcSin[Tan[(c + d*x)/2]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[(1 + Sec[c + d*x ])^(-1)]*Sqrt[1 + Sec[c + d*x]]))/(3840*d*Sqrt[Sec[c + d*x]])
Time = 0.45 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 4375, 374, 25, 27, 441, 27, 445, 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.
| \(\displaystyle \int \cot ^6(c+d x) \sqrt {a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle -\frac {2 \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle -\frac {2 \left (\frac {\int -\frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {9 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {9 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {1}{8} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {9 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )}{a^2 d}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {119 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+87\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {1}{4} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {119 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+87\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{10} \int \frac {5 a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {87 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+23\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {87 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+23\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int -\frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (105-\frac {23 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (105-\frac {23 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {105}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {105 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+361\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {105}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {105 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+361}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {105}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (256 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-151 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )+\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {105}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {151 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {256 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )+\frac {23}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )-\frac {87}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {17 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )}{a^2 d}\) |
Input:
Int[Cot[c + d*x]^6*Sqrt[a + a*Sec[c + d*x]],x]
Output:
(-2*((Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(8*(2 + (a*Tan[c + d*x]^2 )/(a + a*Sec[c + d*x]))^2) + (((-87*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5 /2))/10 + (a*((23*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/6 + (a*(-1/2* (a*((-256*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (151*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]) /(Sqrt[2]*Sqrt[a]))) + (105*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2)) /2)/4 + (17*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(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 = 0.65 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.59
| method | result | size |
| default | \(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (30720 \cos \left (d x +c \right )^{3}+92160 \cos \left (d x +c \right )^{2}+92160 \cos \left (d x +c \right )+30720\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+\left (18120 \cos \left (d x +c \right )^{3}+54360 \cos \left (d x +c \right )^{2}+54360 \cos \left (d x +c \right )+18120\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\sqrt {2}\, \left (20209 \cos \left (d x +c \right )^{7}+112649 \cos \left (d x +c \right )^{6}+107085 \cos \left (d x +c \right )^{5}-112955 \cos \left (d x +c \right )^{4}-189805 \cos \left (d x +c \right )^{3}-8613 \cos \left (d x +c \right )^{2}+87087 \cos \left (d x +c \right )+33495\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}+\left (-4718 \cos \left (d x +c \right )^{7}+53920 \cos \left (d x +c \right )^{6}-29350 \cos \left (d x +c \right )^{5}-100400 \cos \left (d x +c \right )^{4}+34070 \cos \left (d x +c \right )^{3}+83968 \cos \left (d x +c \right )^{2}-12290 \cos \left (d x +c \right )-25200\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{30720 d \left (1+\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right )}\) | \(457\) |
Input:
int(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/30720/d*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^2+2*cos(d*x+ c)+1)*((30720*cos(d*x+c)^3+92160*cos(d*x+c)^2+92160*cos(d*x+c)+30720)*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))+(1812 0*cos(d*x+c)^3+54360*cos(d*x+c)^2+54360*cos(d*x+c)+18120)*(-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)+cs c(d*x+c))+2^(1/2)*(20209*cos(d*x+c)^7+112649*cos(d*x+c)^6+107085*cos(d*x+c )^5-112955*cos(d*x+c)^4-189805*cos(d*x+c)^3-8613*cos(d*x+c)^2+87087*cos(d* x+c)+33495)*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c )))^(1/2)*cot(d*x+c)*csc(d*x+c)^4+(-4718*cos(d*x+c)^7+53920*cos(d*x+c)^6-2 9350*cos(d*x+c)^5-100400*cos(d*x+c)^4+34070*cos(d*x+c)^3+83968*cos(d*x+c)^ 2-12290*cos(d*x+c)-25200)*cot(d*x+c)*csc(d*x+c)^4)
Time = 0.18 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.20 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[1/3840*(2265*sqrt(1/2)*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(-a)*l og(-(4*sqrt(1/2)*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) + 1920*(cos(d*x + c)^4 - 2*cos(d* x + c)^2 + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - co s(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) - 2*(2821*cos(d*x + c)^5 - 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 + 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c)), -1/1920*(2265*sqrt(1/2)*(cos (d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(a)*arctan(2*sqrt(1/2)*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) + 1920*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 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) + (2821*cos(d*x + c)^5 - 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 + 230*cos(d*x + c)^2 + 1575*cos (d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^4 - 2 *d*cos(d*x + c)^2 + d)*sin(d*x + c))]
\[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \cot ^{6}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**6*(a+a*sec(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a*(sec(c + d*x) + 1))*cot(c + d*x)**6, x)
Timed out. \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
Timed out
Time = 0.46 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.65 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
-1/7680*sqrt(2)*(30*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*tan(1/2*d*x + 1 /2*c)^2 - 25)*tan(1/2*d*x + 1/2*c) - 3840*sqrt(2)*sqrt(-a)*a*log(abs(2*(sq rt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 4*s qrt(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) - 2265*sqrt(-a) *log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)) ^2) - 64*(165*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c )^2 + a))^8*sqrt(-a)*a - 555*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan( 1/2*d*x + 1/2*c)^2 + a))^6*sqrt(-a)*a^2 + 785*(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)*a^3 - 505*(sqrt(-a)*t an(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*sqrt(-a)*a^4 + 134*sqrt(-a)*a^5)/((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^5)*sgn(cos(d*x + c))/d
Timed out. \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^6\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \] Input:
int(cot(c + d*x)^6*(a + a/cos(c + d*x))^(1/2),x)
Output:
int(cot(c + d*x)^6*(a + a/cos(c + d*x))^(1/2), x)
\[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{6}d x \right ) \] Input:
int(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**6,x)