Integrand size = 25, antiderivative size = 196 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}+\frac {b (3 a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a f}-\frac {(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f} \]
1/2*(3*a+4*b)*arctanh(b^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))*b^(1/2) /f+1/2*b*(3*a+4*b)*(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)/a/f-1/3*(3*a+4*b)*c ot(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2)/a/f-2/3*cot(f*x+e)^3*(a+b*tan(f*x+e)^2) ^(5/2)/a/f-1/5*cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(5/2)/a/f
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.02 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.09 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-\frac {2 \left (8 a^2+34 a b+3 b^2\right ) \cot (e+f x)}{a}-4 (2 a+3 b) \cot (e+f x) \csc ^2(e+f x)-6 a \cot (e+f x) \csc ^4(e+f x)+\frac {15 \sqrt {2} (3 a+4 b) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right )}{\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}+15 b \tan (e+f x)\right )}{30 \sqrt {2} f} \]
(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*((-2*(8*a^2 + 34* a*b + 3*b^2)*Cot[e + f*x])/a - 4*(2*a + 3*b)*Cot[e + f*x]*Csc[e + f*x]^2 - 6*a*Cot[e + f*x]*Csc[e + f*x]^4 + (15*Sqrt[2]*(3*a + 4*b)*Cot[e + f*x]*El lipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b] /Sqrt[2]], 1])/Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b] + 15*b*Tan[e + f*x]))/(30*Sqrt[2]*f)
Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4146, 365, 27, 359, 247, 211, 224, 219}
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.
| \(\displaystyle \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\sin (e+f x)^6}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \cot ^6(e+f x) \left (\tan ^2(e+f x)+1\right )^2 \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\frac {\int 5 a \cot ^4(e+f x) \left (\tan ^2(e+f x)+2\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{5 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cot ^4(e+f x) \left (\tan ^2(e+f x)+2\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}}{f}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {(3 a+4 b) \int \cot ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{3 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {\frac {(3 a+4 b) \left (3 b \int \sqrt {b \tan ^2(e+f x)+a}d\tan (e+f x)-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {(3 a+4 b) \left (3 b \left (\frac {1}{2} a \int \frac {1}{\sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {(3 a+4 b) \left (3 b \left (\frac {1}{2} a \int \frac {1}{1-\frac {b \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {(3 a+4 b) \left (3 b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 \sqrt {b}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a}-\frac {2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
((-2*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/(3*a) - (Cot[e + f*x]^5* (a + b*Tan[e + f*x]^2)^(5/2))/(5*a) + ((3*a + 4*b)*(-(Cot[e + f*x]*(a + b* Tan[e + f*x]^2)^(3/2)) + 3*b*((a*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b *Tan[e + f*x]^2]])/(2*Sqrt[b]) + (Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2]) /2)))/(3*a))/f
3.2.15.3.1 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)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(768\) vs. \(2(172)=344\).
Time = 6.13 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.92
| method | result | size |
| default | \(-\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (6 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {5}{2}} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )-83 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {3}{2}} a \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{3}+45 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a^{2} b \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{2}+60 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a \,b^{2} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{2}+16 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b}\, a^{2} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{5}-45 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a^{2} b \cos \left (f x +e \right ) \cot \left (f x +e \right )^{2}-60 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a \,b^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )^{2}+110 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {3}{2}} a \cot \left (f x +e \right )^{3}-40 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b}\, a^{2} \cot \left (f x +e \right )^{5}-15 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {3}{2}} a \cot \left (f x +e \right ) \csc \left (f x +e \right )^{2}+30 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b}\, a^{2} \cot \left (f x +e \right )^{3} \csc \left (f x +e \right )^{2}\right )}{30 f a \sqrt {b}\, \left (a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}\right ) \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(769\) |
-1/30/f/a/b^(1/2)*(a+b*tan(f*x+e)^2)^(3/2)/(a*cos(f*x+e)^2+b*sin(f*x+e)^2) /((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*(6*((a*cos(f*x +e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(5/2)*cos(f*x+e)^2*cot(f *x+e)-83*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(3/2 )*a*cos(f*x+e)^2*cot(f*x+e)^3+45*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin( f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a^2*b*cos(f*x+e )^2*cot(f*x+e)^2+60*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(co s(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*b^2*cos(f*x+e)^2*cot(f*x+e )^2+16*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(1/2)* a^2*cos(f*x+e)^2*cot(f*x+e)^5-45*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin( f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a^2*b*cos(f*x+e )*cot(f*x+e)^2-60*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos( f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*b^2*cos(f*x+e)*cot(f*x+e)^2+ 110*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(3/2)*a*c ot(f*x+e)^3-40*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)* b^(1/2)*a^2*cot(f*x+e)^5-15*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e) +1)^2)^(1/2)*b^(3/2)*a*cot(f*x+e)*csc(f*x+e)^2+30*((a*cos(f*x+e)^2-b*cos(f *x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(1/2)*a^2*cot(f*x+e)^3*csc(f*x+e)^2)
Time = 2.73 (sec) , antiderivative size = 655, normalized size of antiderivative = 3.34 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {15 \, {\left ({\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )^{3} + {\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )\right )} \sqrt {b} \log \left (\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) \sin \left (f x + e\right ) - 4 \, {\left ({\left (16 \, a^{2} + 83 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - {\left (40 \, a^{2} + 193 \, a b + 12 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (30 \, a^{2} + 125 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, a b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{120 \, {\left (a f \cos \left (f x + e\right )^{5} - 2 \, a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}, -\frac {15 \, {\left ({\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )^{3} + {\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )\right )} \sqrt {-b} \arctan \left (\frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, {\left ({\left (16 \, a^{2} + 83 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - {\left (40 \, a^{2} + 193 \, a b + 12 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (30 \, a^{2} + 125 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, a b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{60 \, {\left (a f \cos \left (f x + e\right )^{5} - 2 \, a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}\right ] \]
[1/120*(15*((3*a^2 + 4*a*b)*cos(f*x + e)^5 - 2*(3*a^2 + 4*a*b)*cos(f*x + e )^3 + (3*a^2 + 4*a*b)*cos(f*x + e))*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos (f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e )^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((16*a^2 + 83* a*b + 6*b^2)*cos(f*x + e)^6 - (40*a^2 + 193*a*b + 12*b^2)*cos(f*x + e)^4 + (30*a^2 + 125*a*b + 6*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))*sin(f*x + e)), -1/60*(15*((3*a^2 + 4*a*b)*cos(f*x + e) ^5 - 2*(3*a^2 + 4*a*b)*cos(f*x + e)^3 + (3*a^2 + 4*a*b)*cos(f*x + e))*sqrt (-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqr t(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^ 2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*((16*a^2 + 83*a*b + 6*b^2)*cos(f* x + e)^6 - (40*a^2 + 193*a*b + 12*b^2)*cos(f*x + e)^4 + (30*a^2 + 125*a*b + 6*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f* x + e)^2))/((a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)) *sin(f*x + e))]
Timed out. \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.03 \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {45 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 60 \, b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 45 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b \tan \left (f x + e\right ) + \frac {60 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b^{2} \tan \left (f x + e\right )}{a} - \frac {30 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{\tan \left (f x + e\right )} - \frac {40 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b}{a \tan \left (f x + e\right )} - \frac {20 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}{a \tan \left (f x + e\right )^{3}} - \frac {6 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}{a \tan \left (f x + e\right )^{5}}}{30 \, f} \]
1/30*(45*a*sqrt(b)*arcsinh(b*tan(f*x + e)/sqrt(a*b)) + 60*b^(3/2)*arcsinh( b*tan(f*x + e)/sqrt(a*b)) + 45*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e) + 60*sqrt(b*tan(f*x + e)^2 + a)*b^2*tan(f*x + e)/a - 30*(b*tan(f*x + e)^2 + a)^(3/2)/tan(f*x + e) - 40*(b*tan(f*x + e)^2 + a)^(3/2)*b/(a*tan(f*x + e) ) - 20*(b*tan(f*x + e)^2 + a)^(5/2)/(a*tan(f*x + e)^3) - 6*(b*tan(f*x + e) ^2 + a)^(5/2)/(a*tan(f*x + e)^5))/f
\[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{6} \,d x } \]
Timed out. \[ \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^6} \,d x \]