Integrand size = 23, antiderivative size = 259 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 \sqrt {a-b} f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^5 f}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )} \]
-3/8*(a^2-12*a*b+16*b^2)*arctanh(cos(f*x+e))/a^5/f-1/8*(5*a-8*b)*cot(f*x+e )*csc(f*x+e)/a^2/f/(a-b+b*sec(f*x+e)^2)^2-1/4*cot(f*x+e)^3*csc(f*x+e)/a/f/ (a-b+b*sec(f*x+e)^2)^2-1/8*(7*a-12*b)*b*sec(f*x+e)/a^3/f/(a-b+b*sec(f*x+e) ^2)^2-3/2*(a-2*b)*b*sec(f*x+e)/a^4/f/(a-b+b*sec(f*x+e)^2)-3/8*(5*a^2-20*a* b+16*b^2)*arctan(sec(f*x+e)*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^5/f/(a-b)^(1/2)
Time = 7.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}-\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}+\frac {b^2 \cos (e+f x)}{a^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {3 \left (3 a b \cos (e+f x)-4 b^2 \cos (e+f x)\right )}{4 a^4 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}-\frac {3 (a-4 b) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}-\frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 (a-4 b) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}+\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f} \]
(-3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sec[(e + f*x)/2] *(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a ^5*(-a + b)*f) - (3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[( Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]) )/Sqrt[b]])/(8*a^5*(-a + b)*f) + (b^2*Cos[e + f*x])/(a^3*f*(a + b + a*Cos[ 2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (3*(3*a*b*Cos[e + f*x] - 4*b^2*Cos [e + f*x]))/(4*a^4*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) - (3*(a - 4*b)*Csc[(e + f*x)/2]^2)/(32*a^4*f) - Csc[(e + f*x)/2]^4/(64*a^3*f ) - (3*(a^2 - 12*a*b + 16*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^5*f) + (3*(a^2 - 12*a*b + 16*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^5*f) + (3*(a - 4*b)*Sec[(e + f*x)/2]^2)/(32*a^4*f) + Sec[(e + f*x)/2]^4/(64*a^3*f)
Time = 0.53 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4147, 25, 372, 402, 25, 402, 27, 402, 27, 397, 218, 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.
| \(\displaystyle \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^5 \left (a+b \tan (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\int \frac {(4 a-7 b) \sec ^2(e+f x)+a-b}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {(3 a-8 b) (a-b)-5 (5 a-8 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{2 a}+\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\int \frac {(3 a-8 b) (a-b)-5 (5 a-8 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\int -\frac {12 (a-b) \left ((a-4 b) (a-b)-(7 a-12 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{4 a (a-b)}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \int \frac {(a-4 b) (a-b)-(7 a-12 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \left (\frac {4 b (a-2 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}-\frac {\int -\frac {2 (a-b) \left (a^2-8 b a+8 b^2-4 (a-2 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{2 a (a-b)}\right )}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \left (\frac {\int \frac {a^2-8 b a+8 b^2-4 (a-2 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{a}+\frac {4 b (a-2 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2-12 a b+16 b^2\right ) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}+\frac {b \left (5 a^2-20 a b+16 b^2\right ) \int \frac {1}{b \sec ^2(e+f x)+a-b}d\sec (e+f x)}{a}}{a}+\frac {4 b (a-2 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2-12 a b+16 b^2\right ) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}+\frac {\sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{a}+\frac {4 b (a-2 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {(5 a-8 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {\left (a^2-12 a b+16 b^2\right ) \text {arctanh}(\sec (e+f x))}{a}}{a}+\frac {4 b (a-2 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{a}+\frac {b (7 a-12 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )^2}}{f}\) |
(-1/4*Sec[e + f*x]/(a*(1 - Sec[e + f*x]^2)^2*(a - b + b*Sec[e + f*x]^2)^2) + (((5*a - 8*b)*Sec[e + f*x])/(2*a*(1 - Sec[e + f*x]^2)*(a - b + b*Sec[e + f*x]^2)^2) - (((7*a - 12*b)*b*Sec[e + f*x])/(a*(a - b + b*Sec[e + f*x]^2 )^2) + (3*(((Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x ])/Sqrt[a - b]])/(a*Sqrt[a - b]) + ((a^2 - 12*a*b + 16*b^2)*ArcTanh[Sec[e + f*x]])/a)/a + (4*(a - 2*b)*b*Sec[e + f*x])/(a*(a - b + b*Sec[e + f*x]^2) )))/a)/(2*a))/(4*a))/f
3.1.85.3.1 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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -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]
Time = 3.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}+\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}}{f}\) | \(265\) |
| default | \(\frac {\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}+\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}}{f}\) | \(265\) |
| risch | \(\text {Expression too large to display}\) | \(1036\) |
1/f*(b/a^5*((-3/8*a*(3*a^2-7*a*b+4*b^2)*cos(f*x+e)^3+(-7/8*a^2*b+3/2*a*b^2 )*cos(f*x+e))/(a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)^2+3/8*(5*a^2-20*a*b+16*b^2 )/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2)))+1/16/a^3/(cos( f*x+e)+1)^2-1/16*(-3*a+12*b)/a^4/(cos(f*x+e)+1)+1/16/a^5*(-3*a^2+36*a*b-48 *b^2)*ln(cos(f*x+e)+1)-1/16/a^3/(cos(f*x+e)-1)^2-1/16*(-3*a+12*b)/a^4/(cos (f*x+e)-1)+1/16/a^5*(3*a^2-36*a*b+48*b^2)*ln(cos(f*x+e)-1))
Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (239) = 478\).
Time = 0.46 (sec) , antiderivative size = 1693, normalized size of antiderivative = 6.54 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
[1/16*(6*(a^4 - 9*a^3*b + 16*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 46*a^3*b + 108*a^2*b^2 - 72*a*b^3)*cos(f*x + e)^5 - 2*(19*a^3*b - 72*a^2 *b^2 + 72*a*b^3)*cos(f*x + e)^3 + 3*((5*a^4 - 30*a^3*b + 61*a^2*b^2 - 52*a *b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(5*a^4 - 35*a^3*b + 86*a^2*b^2 - 88*a*b^ 3 + 32*b^4)*cos(f*x + e)^6 + (5*a^4 - 50*a^3*b + 166*a^2*b^2 - 216*a*b^3 + 96*b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 20*a*b^3 + 16*b^4 + 2*(5*a^3*b - 30* a^2*b^2 + 56*a*b^3 - 32*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(((a - b) *cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*co s(f*x + e)^2 + b)) - 24*(a^2*b^2 - 2*a*b^3)*cos(f*x + e) - 3*((a^4 - 14*a^ 3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2 *b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f* x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e )^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*c os(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 2*a^6*b + a^5*b^2)*f* cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^ 6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2...
Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (239) = 478\).
Time = 0.91 (sec) , antiderivative size = 861, normalized size of antiderivative = 3.32 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
1/64*(12*(a^2 - 12*a*b + 16*b^2)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/a^5 - 24*(5*a^2*b - 20*a*b^2 + 16*b^3)*arctan(-(a*cos(f*x + e) - b*cos(f*x + e) - b)/(sqrt(a*b - b^2)*cos(f*x + e) + sqrt(a*b - b^2)))/(sqr t(a*b - b^2)*a^5) - (8*a^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 24*a^2* b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - a^3*(cos(f*x + e) - 1)^2/(cos(f* x + e) + 1)^2)/a^6 - (a^4 - 4*a^4*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 16*a^3*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 20*a^4*(cos(f*x + e) - 1) ^2/(cos(f*x + e) + 1)^2 + 216*a^3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1 )^2 - 304*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 20*a^4*(cos( f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 360*a^3*b*(cos(f*x + e) - 1)^3/(cos (f*x + e) + 1)^3 - 1024*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 896*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 5*a^4*(cos(f*x + e ) - 1)^4/(cos(f*x + e) + 1)^4 + 64*a^3*b*(cos(f*x + e) - 1)^4/(cos(f*x + e ) + 1)^4 - 192*a^2*b^2*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 + 256*a*b ^3*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 - 256*b^4*(cos(f*x + e) - 1)^ 4/(cos(f*x + e) + 1)^4 + 16*a^4*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 168*a^3*b*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 + 384*a^2*b^2*(cos(f *x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 256*a*b^3*(cos(f*x + e) - 1)^5/(cos( f*x + e) + 1)^5 + 6*a^4*(cos(f*x + e) - 1)^6/(cos(f*x + e) + 1)^6 - 72*a^3 *b*(cos(f*x + e) - 1)^6/(cos(f*x + e) + 1)^6 + 96*a^2*b^2*(cos(f*x + e)...
Time = 11.94 (sec) , antiderivative size = 1357, normalized size of antiderivative = 5.24 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
tan(e/2 + (f*x)/2)^4/(64*a^3*f) + (tan(e/2 + (f*x)/2)^2*((3*(a - 2*b))/(16 *a^4) - 1/(16*a^3)))/f + (tan(e/2 + (f*x)/2)^4*(100*a*b^2 - 72*a^2*b + (13 *a^3)/2) - tan(e/2 + (f*x)/2)^10*(144*a*b^2 - 42*a^2*b + 2*a^3 - 128*b^3) - tan(e/2 + (f*x)/2)^6*(496*a*b^2 - 174*a^2*b + 11*a^3 - 416*b^3) + tan(e/ 2 + (f*x)/2)^2*(4*a^2*b - a^3) - a^3/4 + (tan(e/2 + (f*x)/2)^8*(31*a^4 - 5 92*a^3*b - 2944*a*b^3 + 1792*b^4 + 2016*a^2*b^2))/(4*a))/(f*(16*a^6*tan(e/ 2 + (f*x)/2)^4 + 16*a^6*tan(e/2 + (f*x)/2)^12 + tan(e/2 + (f*x)/2)^8*(96*a ^6 - 256*a^5*b + 256*a^4*b^2) + tan(e/2 + (f*x)/2)^6*(128*a^5*b - 64*a^6) + tan(e/2 + (f*x)/2)^10*(128*a^5*b - 64*a^6))) + (log(tan(e/2 + (f*x)/2))* (3*a^2 - 36*a*b + 48*b^2))/(8*a^5*f) + (3*b^(1/2)*atan((a^13*(a - b)^(3/2) *((256*((3456*b^8 - 11232*a*b^7 + 14256*a^2*b^6 - 8910*a^3*b^5 + 2835*a^4* b^4 - (3375*a^5*b^3)/8 + (675*a^6*b^2)/32)/a^12 - (9*b*(5*a^2 - 20*a*b + 1 6*b^2)^2*(192*a^14 - 4992*a^13*b + 24576*a^10*b^4 - 43008*a^11*b^3 + 24576 *a^12*b^2))/(8192*a^22*(a - b)))*(1728*a*b^4 - 45*a^4*b + a^5 - 768*b^5 - 1344*a^2*b^3 + 420*a^3*b^2))/(b^(1/2)*(b*(b*(b*(1680*a^7 + b*(768*a^5*b - 1920*a^6)) - 600*a^8) + 75*a^9) - 4*a^10)) - 256*tan(e/2 + (f*x)/2)^2*(((( 4752*a*b^6 - 1728*b^7 - 4860*a^2*b^5 + 2295*a^3*b^4 - (2025*a^4*b^3)/4 + ( 675*a^5*b^2)/16)/a^11 + (9*b*(5*a^2 - 20*a*b + 16*b^2)^2*(3552*a^12*b - 96 *a^13 + 73728*a^8*b^5 - 165888*a^9*b^4 + 125952*a^10*b^3 - 36480*a^11*b^2) )/(4096*a^21*(a - b)))*(1728*a*b^4 - 45*a^4*b + a^5 - 768*b^5 - 1344*a^...