Integrand size = 21, antiderivative size = 65 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac {4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \] Output:
-2/11*d^5/b/(d*tan(b*x+a))^(11/2)-4/7*d^3/b/(d*tan(b*x+a))^(7/2)-2/3*d/b/( d*tan(b*x+a))^(3/2)
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {2 d (-45+28 \cos (2 (a+b x))-4 \cos (4 (a+b x))) \csc ^4(a+b x)}{231 b (d \tan (a+b x))^{3/2}} \] Input:
Integrate[Csc[a + b*x]^6/Sqrt[d*Tan[a + b*x]],x]
Output:
(2*d*(-45 + 28*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(231 *b*(d*Tan[a + b*x])^(3/2))
Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3071, 244, 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.
\(\displaystyle \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (a+b x)^6 \sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3071 |
\(\displaystyle \frac {d \int \frac {\left (\tan ^2(a+b x) d^2+d^2\right )^2}{(d \tan (a+b x))^{13/2}}d(d \tan (a+b x))}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {d \int \left (\frac {d^4}{(d \tan (a+b x))^{13/2}}+\frac {2 d^2}{(d \tan (a+b x))^{9/2}}+\frac {1}{(d \tan (a+b x))^{5/2}}\right )d(d \tan (a+b x))}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d \left (-\frac {2 d^4}{11 (d \tan (a+b x))^{11/2}}-\frac {4 d^2}{7 (d \tan (a+b x))^{7/2}}-\frac {2}{3 (d \tan (a+b x))^{3/2}}\right )}{b}\) |
Input:
Int[Csc[a + b*x]^6/Sqrt[d*Tan[a + b*x]],x]
Output:
(d*((-2*d^4)/(11*(d*Tan[a + b*x])^(11/2)) - (4*d^2)/(7*(d*Tan[a + b*x])^(7 /2)) - 2/(3*(d*Tan[a + b*x])^(3/2))))/b
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Time = 0.78 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65
method | result | size |
default | \(-\frac {\cot \left (b x +a \right ) \csc \left (b x +a \right )^{4} \left (32 \cos \left (b x +a \right )^{4}-88 \cos \left (b x +a \right )^{2}+77\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}}{231 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \tan \left (b x +a \right )}}\) | \(107\) |
Input:
int(csc(b*x+a)^6/(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/231/b*cot(b*x+a)*csc(b*x+a)^4*(32*cos(b*x+a)^4-88*cos(b*x+a)^2+77)*(-2* sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(-sin(b*x+a)*cos(b*x+a)/(cos (b*x+a)+1)^2)^(1/2)/(d*tan(b*x+a))^(1/2)*2^(1/2)
Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{6} - 88 \, \cos \left (b x + a\right )^{4} + 77 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{231 \, {\left (b d \cos \left (b x + a\right )^{6} - 3 \, b d \cos \left (b x + a\right )^{4} + 3 \, b d \cos \left (b x + a\right )^{2} - b d\right )}} \] Input:
integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(1/2),x, algorithm="fricas")
Output:
2/231*(32*cos(b*x + a)^6 - 88*cos(b*x + a)^4 + 77*cos(b*x + a)^2)*sqrt(d*s in(b*x + a)/cos(b*x + a))/(b*d*cos(b*x + a)^6 - 3*b*d*cos(b*x + a)^4 + 3*b *d*cos(b*x + a)^2 - b*d)
\[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\csc ^{6}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \] Input:
integrate(csc(b*x+a)**6/(d*tan(b*x+a))**(1/2),x)
Output:
Integral(csc(a + b*x)**6/sqrt(d*tan(a + b*x)), x)
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \, {\left (77 \, d^{4} \tan \left (b x + a\right )^{4} + 66 \, d^{4} \tan \left (b x + a\right )^{2} + 21 \, d^{4}\right )} d}{231 \, \left (d \tan \left (b x + a\right )\right )^{\frac {11}{2}} b} \] Input:
integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
Output:
-2/231*(77*d^4*tan(b*x + a)^4 + 66*d^4*tan(b*x + a)^2 + 21*d^4)*d/((d*tan( b*x + a))^(11/2)*b)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \, {\left (77 \, d^{5} \tan \left (b x + a\right )^{4} + 66 \, d^{5} \tan \left (b x + a\right )^{2} + 21 \, d^{5}\right )}}{231 \, \sqrt {d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{5}} \] Input:
integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(1/2),x, algorithm="giac")
Output:
-2/231*(77*d^5*tan(b*x + a)^4 + 66*d^5*tan(b*x + a)^2 + 21*d^5)/(sqrt(d*ta n(b*x + a))*b*d^5*tan(b*x + a)^5)
Time = 8.36 (sec) , antiderivative size = 831, normalized size of antiderivative = 12.78 \[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\text {Too large to display} \] Input:
int(1/(sin(a + b*x)^6*(d*tan(a + b*x))^(1/2)),x)
Output:
((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b *x*2i) + 1))^(1/2)*44864i)/(10395*b*d*(exp(a*2i + b*x*2i)*1i - 1i)) - (128 *(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b *x*2i) + 1))^(1/2))/(35*b*d*(exp(a*2i + b*x*2i) - 1)^2) - (7136*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1) )^(1/2))/(1155*b*d*(exp(a*2i + b*x*2i) - 1)^3) - (1216*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/ (231*b*d*(exp(a*2i + b*x*2i) - 1)^4) - (160*(exp(a*2i + b*x*2i) + 1)*(-(d* (exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(99*b*d*(ex p(a*2i + b*x*2i) - 1)^5) - (41984*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(10395*b*d*(exp(a*2i + b*x*2i) - 1)) - (3904*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)* 1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(1155*b*d*(exp(a*2i + b*x*2i)*1 i - 1i)^2) - ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/ (exp(a*2i + b*x*2i) + 1))^(1/2)*1088i)/(165*b*d*(exp(a*2i + b*x*2i)*1i - 1 i)^3) + (320*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/( exp(a*2i + b*x*2i) + 1))^(1/2))/(21*b*d*(exp(a*2i + b*x*2i)*1i - 1i)^4) + ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b *x*2i) + 1))^(1/2)*1600i)/(99*b*d*(exp(a*2i + b*x*2i)*1i - 1i)^5) - (64*(e xp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b...
\[ \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (b x +a \right )}\, \csc \left (b x +a \right )^{6}}{\tan \left (b x +a \right )}d x \right )}{d} \] Input:
int(csc(b*x+a)^6/(d*tan(b*x+a))^(1/2),x)
Output:
(sqrt(d)*int((sqrt(tan(a + b*x))*csc(a + b*x)**6)/tan(a + b*x),x))/d