Integrand size = 25, antiderivative size = 223 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {3 \left (a^2+6 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f} \]
-1/4*cot(f*x+e)*csc(f*x+e)^3*(a-b+b*sec(f*x+e)^2)^(3/2)/f-3/8*(a^2+6*a*b+b ^2)*arctanh(sec(f*x+e)*a^(1/2)/(a-b+b*sec(f*x+e)^2)^(1/2))/f/a^(1/2)+3/2*( a+b)*arctanh(sec(f*x+e)*b^(1/2)/(a-b+b*sec(f*x+e)^2)^(1/2))*b^(1/2)/f+3/8* (a+3*b)*sec(f*x+e)*(a-b+b*sec(f*x+e)^2)^(1/2)/f-3/8*(a+b)*csc(f*x+e)^2*sec (f*x+e)*(a-b+b*sec(f*x+e)^2)^(1/2)/f
Time = 5.53 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.86 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\cos (e+f x) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-2 \csc ^2(e+f x) \left (3 a+5 b+2 a \csc ^2(e+f x)\right )+8 b \sec ^2(e+f x)+\frac {3 \left (16 \sqrt {a} \sqrt {b} (a+b) \text {arctanh}\left (\frac {-\sqrt {a} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{2 \sqrt {b}}\right )+\left (a^2+6 a b+b^2\right ) \left (2 \text {arctanh}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\cos ^2(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {a} \sqrt {\left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}\right )}{16 \sqrt {2} f} \]
(Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-2* Csc[e + f*x]^2*(3*a + 5*b + 2*a*Csc[e + f*x]^2) + 8*b*Sec[e + f*x]^2 + (3* (16*Sqrt[a]*Sqrt[b]*(a + b)*ArcTanh[(-(Sqrt[a]*(-1 + Tan[(e + f*x)/2]^2)) + Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])/(2*Sqrt[b] )] + (a^2 + 6*a*b + b^2)*(2*ArcTanh[Tan[(e + f*x)/2]^2 - Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]/Sqrt[a]] + Log[a - 2*b - a*Tan [(e + f*x)/2]^2 + Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f *x)/2]^2)^2]]))*Sec[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]^2*Sec[(e + f*x)/2]^4] )/(Sqrt[a]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])))/(16*Sqrt[2]*f)
Time = 0.49 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4147, 25, 369, 27, 439, 25, 444, 27, 398, 224, 219, 291, 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.
| \(\displaystyle \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\sin (e+f x)^5}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\frac {\sec ^4(e+f x) \left (b \sec ^2(e+f x)+a-b\right )^{3/2}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sec ^4(e+f x) \left (b \sec ^2(e+f x)+a-b\right )^{3/2}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {3 \sec ^2(e+f x) \sqrt {b \sec ^2(e+f x)+a-b} \left (2 b \sec ^2(e+f x)+a-b\right )}{\left (1-\sec ^2(e+f x)\right )^2}d\sec (e+f x)-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{4} \int \frac {\sec ^2(e+f x) \sqrt {b \sec ^2(e+f x)+a-b} \left (2 b \sec ^2(e+f x)+a-b\right )}{\left (1-\sec ^2(e+f x)\right )^2}d\sec (e+f x)-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int -\frac {\sec ^2(e+f x) \left (2 b (a+3 b) \sec ^2(e+f x)+(a-b) (a+5 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}-\frac {1}{2} \int \frac {\sec ^2(e+f x) \left (2 b (a+3 b) \sec ^2(e+f x)+(a-b) (a+5 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left ((a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}-\frac {\int \frac {2 b \left (4 b (a+b) \sec ^2(e+f x)+(a-b) (a+3 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)}{2 b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left ((a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}-\int \frac {4 b (a+b) \sec ^2(e+f x)+(a-b) (a+3 b)}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 b (a+b) \int \frac {1}{\sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 b (a+b) \int \frac {1}{1-\frac {b \sec ^2(e+f x)}{b \sec ^2(e+f x)+a-b}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a-b}}+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{1-\frac {a \sec ^2(e+f x)}{b \sec ^2(e+f x)+a-b}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a-b}}+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\frac {\left (a^2+6 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{\sqrt {a}}+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\) |
(-1/4*(Sec[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(1 - Sec[e + f*x]^ 2)^2 + (3*(((a + b)*Sec[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(1 - Sec[e + f*x]^2)) + (-(((a^2 + 6*a*b + b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x]) /Sqrt[a - b + b*Sec[e + f*x]^2]])/Sqrt[a]) + 4*Sqrt[b]*(a + b)*ArcTanh[(Sq rt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]] + (a + 3*b)*Sec[e + f* x]*Sqrt[a - b + b*Sec[e + f*x]^2])/2))/4)/f
3.2.9.3.1 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]))* 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[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/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(2517\) vs. \(2(195)=390\).
Time = 2.56 (sec) , antiderivative size = 2518, normalized size of antiderivative = 11.29
1/16/f/b/a^(5/2)*(-24*sin(f*x+e)^2*cos(f*x+e)^3*ln(-4*b^(1/2)*((a*cos(f*x+ e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)-4*b^(1/2)*((a*cos(f*x+e)^2- b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*sec(f*x+e)-4*b*sec(f*x+e))*a^(7/ 2)*b^(3/2)-24*sin(f*x+e)^2*cos(f*x+e)^3*ln(-4*b^(1/2)*((a*cos(f*x+e)^2-b*c os(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)-4*b^(1/2)*((a*cos(f*x+e)^2-b*cos(f* x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*sec(f*x+e)-4*b*sec(f*x+e))*a^(5/2)*b^(5/ 2)+24*sin(f*x+e)^2*cos(f*x+e)^2*ln(-4*b^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e )^2+b)/(cos(f*x+e)+1)^2)^(1/2)-4*b^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b )/(cos(f*x+e)+1)^2)^(1/2)*sec(f*x+e)-4*b*sec(f*x+e))*a^(7/2)*b^(3/2)+24*si n(f*x+e)^2*cos(f*x+e)^2*ln(-4*b^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/( cos(f*x+e)+1)^2)^(1/2)-4*b^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f *x+e)+1)^2)^(1/2)*sec(f*x+e)-4*b*sec(f*x+e))*a^(5/2)*b^(5/2)+6*cos(f*x+e)^ 4*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a^(7/2)*b-18* sin(f*x+e)^2*cos(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)*a^(5/2)*b^2+3*sin(f*x+e)^2*cos(f*x+e)^3*ln(2/a^(1/2)*(cos(f*x+e )*a^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)+((a*c os(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)-cos(f*x+e)*a +b*cos(f*x+e)+b)/(cos(f*x+e)+1))*a^4*b+18*sin(f*x+e)^2*cos(f*x+e)^3*ln(2/a ^(1/2)*(cos(f*x+e)*a^(1/2)*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+ 1)^2)^(1/2)+((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*...
Time = 1.22 (sec) , antiderivative size = 1365, normalized size of antiderivative = 6.12 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/16*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f *x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos( f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*c os(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 12*((a^2 + a*b)*cos(f*x + e)^ 5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(b)*log(- ((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos( f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(3*(a^2 + 3*a*b)*cos(f *x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*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)), 1/8*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(-a)*a rctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + 6*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a ^2 + a*b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*s qrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f *x + e)^2) + (3*(a^2 + 3*a*b)*cos(f*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*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)), -1/16*(24*((a^2 + a *b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x ...
Timed out. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{5} \,d x } \]
Exception generated. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^5} \,d x \]