Integrand size = 21, antiderivative size = 63 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac {4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac {2 d}{b \sqrt {d \tan (a+b x)}} \] Output:
-2/9*d^5/b/(d*tan(b*x+a))^(9/2)-4/5*d^3/b/(d*tan(b*x+a))^(5/2)-2*d/b/(d*ta n(b*x+a))^(1/2)
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {2 d (-21+20 \cos (2 (a+b x))-4 \cos (4 (a+b x))) \csc ^4(a+b x)}{45 b \sqrt {d \tan (a+b x)}} \] Input:
Integrate[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]
Output:
(2*d*(-21 + 20*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(45* b*Sqrt[d*Tan[a + b*x]])
Time = 0.22 (sec) , antiderivative size = 58, 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.
\(\displaystyle \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {d \tan (a+b x)}}{\sin (a+b x)^6}dx\) |
\(\Big \downarrow \) 3071 |
\(\displaystyle \frac {d \int \frac {\left (\tan ^2(a+b x) d^2+d^2\right )^2}{(d \tan (a+b x))^{11/2}}d(d \tan (a+b x))}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {d \int \left (\frac {d^4}{(d \tan (a+b x))^{11/2}}+\frac {2 d^2}{(d \tan (a+b x))^{7/2}}+\frac {1}{(d \tan (a+b x))^{3/2}}\right )d(d \tan (a+b x))}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d \left (-\frac {2 d^4}{9 (d \tan (a+b x))^{9/2}}-\frac {4 d^2}{5 (d \tan (a+b x))^{5/2}}-\frac {2}{\sqrt {d \tan (a+b x)}}\right )}{b}\) |
Input:
Int[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]
Output:
(d*((-2*d^4)/(9*(d*Tan[a + b*x])^(9/2)) - (4*d^2)/(5*(d*Tan[a + b*x])^(5/2 )) - 2/Sqrt[d*Tan[a + b*x]]))/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 = 0.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {2 \cot \left (b x +a \right ) \csc \left (b x +a \right )^{4} \left (32 \cos \left (b x +a \right )^{4}-72 \cos \left (b x +a \right )^{2}+45\right ) \sqrt {d \tan \left (b x +a \right )}}{45 b}\) | \(52\) |
Input:
int(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/45/b*cot(b*x+a)*csc(b*x+a)^4*(32*cos(b*x+a)^4-72*cos(b*x+a)^2+45)*(d*ta n(b*x+a))^(1/2)
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{5} - 72 \, \cos \left (b x + a\right )^{3} + 45 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{45 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \] Input:
integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")
Output:
-2/45*(32*cos(b*x + a)^5 - 72*cos(b*x + a)^3 + 45*cos(b*x + a))*sqrt(d*sin (b*x + a)/cos(b*x + a))/((b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)*sin(b *x + a))
Timed out. \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**6*(d*tan(b*x+a))**(1/2),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 \, {\left (45 \, d^{4} \tan \left (b x + a\right )^{4} + 18 \, d^{4} \tan \left (b x + a\right )^{2} + 5 \, d^{4}\right )} d}{45 \, \left (d \tan \left (b x + a\right )\right )^{\frac {9}{2}} b} \] Input:
integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
Output:
-2/45*(45*d^4*tan(b*x + a)^4 + 18*d^4*tan(b*x + a)^2 + 5*d^4)*d/((d*tan(b* x + a))^(9/2)*b)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 \, {\left (45 \, d^{6} \tan \left (b x + a\right )^{4} + 18 \, d^{6} \tan \left (b x + a\right )^{2} + 5 \, d^{6}\right )}}{45 \, \sqrt {d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{4}} \] Input:
integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="giac")
Output:
-2/45*(45*d^6*tan(b*x + a)^4 + 18*d^6*tan(b*x + a)^2 + 5*d^6)/(sqrt(d*tan( b*x + a))*b*d^5*tan(b*x + a)^4)
Time = 4.71 (sec) , antiderivative size = 356, normalized size of antiderivative = 5.65 \[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{45\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{45\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{15\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{9\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^4}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{9\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^5} \] Input:
int((d*tan(a + b*x))^(1/2)/sin(a + b*x)^6,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)*64i)/(45*b*(exp(a*2i + b*x*2i) - 1)^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)*64i)/(45*b*(exp(a*2i + b*x*2i) - 1)) - ((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)*32i)/(15*b* (exp(a*2i + b*x*2i) - 1)^3) - ((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)*64i)/(9*b*(exp(a*2i + b*x *2i) - 1)^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)*32i)/(9*b*(exp(a*2i + b*x*2i) - 1)^5)
\[ \int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \csc \left (b x +a \right )^{6}d x \right ) \] 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,x)