Integrand size = 23, antiderivative size = 210 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^4 f}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^4 f}-\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {3 (3 a-4 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )} \]
-3/8*(a^2-8*a*b+8*b^2)*arctanh(cos(f*x+e))/a^4/f-1/8*(5*a-6*b)*cot(f*x+e)* csc(f*x+e)/a^2/f/(a-b+b*sec(f*x+e)^2)-1/4*cot(f*x+e)^3*csc(f*x+e)/a/f/(a-b +b*sec(f*x+e)^2)-3/8*(3*a-4*b)*b*sec(f*x+e)/a^3/f/(a-b+b*sec(f*x+e)^2)-3/2 *(a-2*b)*arctan(sec(f*x+e)*b^(1/2)/(a-b)^(1/2))*(a-b)^(1/2)*b^(1/2)/a^4/f
Time = 7.03 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.87 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^4 f}+\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^4 f}+\frac {-a b \cos (e+f x)+b^2 \cos (e+f x)}{a^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}+\frac {(-3 a+8 b) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^3 f}-\frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^2 f}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^4 f}+\frac {3 \left (a^2-8 a b+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^4 f}+\frac {(3 a-8 b) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^3 f}+\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^2 f} \]
(3*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos [(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(2*a^4*f) + (3*(a - 2 *b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x )/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(2*a^4*f) + (-(a*b*Cos[e + f*x ]) + b^2*Cos[e + f*x])/(a^3*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f *x)])) + ((-3*a + 8*b)*Csc[(e + f*x)/2]^2)/(32*a^3*f) - Csc[(e + f*x)/2]^4 /(64*a^2*f) - (3*(a^2 - 8*a*b + 8*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^4*f) + (3*(a^2 - 8*a*b + 8*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^4*f) + ((3*a - 8*b)*S ec[(e + f*x)/2]^2)/(32*a^3*f) + Sec[(e + f*x)/2]^4/(64*a^2*f)
Time = 0.46 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4147, 25, 372, 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.
| \(\displaystyle \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^5 \left (a+b \tan (e+f x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\int \frac {(4 a-5 b) \sec ^2(e+f x)+a-b}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {3 \left ((a-2 b) (a-b)-(5 a-6 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{2 a}+\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \int \frac {(a-2 b) (a-b)-(5 a-6 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (\frac {b (3 a-4 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}-\frac {\int -\frac {2 (a-b) \left ((a-4 b) (a-b)-(3 a-4 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{2 a (a-b)}\right )}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (\frac {\int \frac {(a-4 b) (a-b)-(3 a-4 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{a}+\frac {b (3 a-4 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (\frac {\frac {\left (a^2-8 a b+8 b^2\right ) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}+\frac {4 b (a-2 b) (a-b) \int \frac {1}{b \sec ^2(e+f x)+a-b}d\sec (e+f x)}{a}}{a}+\frac {b (3 a-4 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (\frac {\frac {\left (a^2-8 a b+8 b^2\right ) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}+\frac {4 \sqrt {b} (a-2 b) \sqrt {a-b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a}}{a}+\frac {b (3 a-4 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {(5 a-6 b) \sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (\frac {\frac {\left (a^2-8 a b+8 b^2\right ) \text {arctanh}(\sec (e+f x))}{a}+\frac {4 \sqrt {b} (a-2 b) \sqrt {a-b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a}}{a}+\frac {b (3 a-4 b) \sec (e+f x)}{a \left (a+b \sec ^2(e+f x)-b\right )}\right )}{2 a}}{4 a}-\frac {\sec (e+f x)}{4 a \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)-b\right )}}{f}\) |
(-1/4*Sec[e + f*x]/(a*(1 - Sec[e + f*x]^2)^2*(a - b + b*Sec[e + f*x]^2)) + (((5*a - 6*b)*Sec[e + f*x])/(2*a*(1 - Sec[e + f*x]^2)*(a - b + b*Sec[e + f*x]^2)) - (3*(((4*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f *x])/Sqrt[a - b]])/a + ((a^2 - 8*a*b + 8*b^2)*ArcTanh[Sec[e + f*x]])/a)/a + ((3*a - 4*b)*b*Sec[e + f*x])/(a*(a - b + b*Sec[e + f*x]^2))))/(2*a))/(4* a))/f
3.1.73.3.1 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 = 1.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {\left (-\frac {1}{2} a^{2}+\frac {1}{2} a b \right ) \cos \left (f x +e \right )}{a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}+\frac {3 \left (a^{2}-3 a b +2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+24 a b -24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{4}}-\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-24 a b +24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{4}}}{f}\) | \(232\) |
| default | \(\frac {\frac {b \left (\frac {\left (-\frac {1}{2} a^{2}+\frac {1}{2} a b \right ) \cos \left (f x +e \right )}{a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}+\frac {3 \left (a^{2}-3 a b +2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+24 a b -24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{4}}-\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-24 a b +24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{4}}}{f}\) | \(232\) |
| risch | \(\frac {3 a^{2} {\mathrm e}^{11 i \left (f x +e \right )}-15 a b \,{\mathrm e}^{11 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{11 i \left (f x +e \right )}-5 a^{2} {\mathrm e}^{9 i \left (f x +e \right )}+21 a b \,{\mathrm e}^{9 i \left (f x +e \right )}-36 b^{2} {\mathrm e}^{9 i \left (f x +e \right )}-30 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}-6 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+24 b^{2} {\mathrm e}^{7 i \left (f x +e \right )}-30 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}-6 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+24 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}-5 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}+21 a b \,{\mathrm e}^{3 i \left (f x +e \right )}-36 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+3 a^{2} {\mathrm e}^{i \left (f x +e \right )}-15 a b \,{\mathrm e}^{i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{i \left (f x +e \right )}}{4 f \,a^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 a^{2} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b}{a^{3} f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{a^{4} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 a^{2} f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b}{a^{3} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{a^{4} f}+\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 f \,a^{3}}-\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{2 f \,a^{4}}-\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 f \,a^{3}}+\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{2 f \,a^{4}}\) | \(702\) |
1/f*(b/a^4*((-1/2*a^2+1/2*a*b)*cos(f*x+e)/(a*cos(f*x+e)^2-b*cos(f*x+e)^2+b )+3/2*(a^2-3*a*b+2*b^2)/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^ (1/2)))+1/16/a^2/(cos(f*x+e)+1)^2-1/16*(-3*a+8*b)/a^3/(cos(f*x+e)+1)+1/16/ a^4*(-3*a^2+24*a*b-24*b^2)*ln(cos(f*x+e)+1)-1/16/a^2/(cos(f*x+e)-1)^2-1/16 *(-3*a+8*b)/a^3/(cos(f*x+e)-1)+1/16/a^4*(3*a^2-24*a*b+24*b^2)*ln(cos(f*x+e )-1))
Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (192) = 384\).
Time = 0.38 (sec) , antiderivative size = 1052, normalized size of antiderivative = 5.01 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
[1/16*(6*(a^3 - 5*a^2*b + 4*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 24*a^2*b + 24*a*b^2)*cos(f*x + e)^3 - 12*((a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^6 - (2*a ^2 - 7*a*b + 6*b^2)*cos(f*x + e)^4 + (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 + a*b - 2*b^2)*sqrt(-a*b + b^2)*log(((a - b)*cos(f*x + e)^2 - 2*sqrt(-a*b + b^2)*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 6*(3*a^2*b - 4*a* b^2)*cos(f*x + e) - 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - (2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8*b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(1/2*cos( f*x + e) + 1/2) + 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - ( 2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8 *b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f *x + e) + 1/2))/((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4 *b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2), 1/16*(6*(a^3 - 5 *a^2*b + 4*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 24*a^2*b + 24*a*b^2)*cos(f*x + e)^3 + 24*((a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^6 - (2*a^2 - 7*a*b + 6*b^ 2)*cos(f*x + e)^4 + (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 + a*b - 2*b^2)*sq rt(a*b - b^2)*arctan(sqrt(a*b - b^2)*cos(f*x + e)/b) - 6*(3*a^2*b - 4*a*b^ 2)*cos(f*x + e) - 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - ( 2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8 *b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(1/2*cos...
Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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. 516 vs. \(2 (192) = 384\).
Time = 0.59 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.46 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {12 \, {\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{4}} - \frac {96 \, {\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{4}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{4}} - \frac {{\left (a^{2} - \frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {144 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {64 \, {\left (a^{2} b - a b^{2} + \frac {a^{2} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {3 \, a b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )}}{{\left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} a^{4}}}{64 \, f} \]
1/64*(12*(a^2 - 8*a*b + 8*b^2)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/a^4 - 96*(a^2*b - 3*a*b^2 + 2*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)))/(sqrt(a*b - b^2)*a^4) - (8*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 16*a*b*(cos(f *x + e) - 1)/(cos(f*x + e) + 1) - a^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/a^4 - (a^2 - 8*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 16*a*b*( cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 18*a^2*(cos(f*x + e) - 1)^2/(cos(f* x + e) + 1)^2 - 144*a*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 144*b^ 2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)^2/(a^4*(co s(f*x + e) - 1)^2) - 64*(a^2*b - a*b^2 + a^2*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 3*a*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 2*b^3*(cos(f* x + e) - 1)/(cos(f*x + e) + 1))/((a + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 4*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a*(cos(f*x + e) - 1)^2 /(cos(f*x + e) + 1)^2)*a^4))/f
Time = 11.14 (sec) , antiderivative size = 1113, normalized size of antiderivative = 5.30 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
tan(e/2 + (f*x)/2)^4/(64*a^2*f) - (a^2/4 - tan(e/2 + (f*x)/2)^4*((15*a^2)/ 4 - 32*a*b + 32*b^2) + (3*a*tan(e/2 + (f*x)/2)^2*(a - 2*b))/2 + (2*tan(e/2 + (f*x)/2)^6*(24*a*b^2 - 10*a^2*b + a^3 - 16*b^3))/a)/(f*(16*a^4*tan(e/2 + (f*x)/2)^4 + 16*a^4*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^6*(64*a^3* b - 32*a^4))) + (tan(e/2 + (f*x)/2)^2*(a - 2*b))/(8*a^3*f) + (log(tan(e/2 + (f*x)/2))*(3*a^2 - 24*a*b + 24*b^2))/(8*a^4*f) + (3*atan((8*a^10*tan(e/2 + (f*x)/2)^2*((((756*a*b^6 - 216*b^7 - 1026*a^2*b^5 + 675*a^3*b^4 - 216*a ^4*b^3 + 27*a^5*b^2)/a^8 + (9*(a - 2*b)^2*(a*b - b^2)*(180*a^10*b - 6*a^11 + 2304*a^6*b^5 - 5760*a^7*b^4 + 4944*a^8*b^3 - 1656*a^9*b^2))/(16*a^16))* (960*a*b^4 - 38*a^4*b + a^5 - 384*b^5 - 840*a^2*b^3 + 300*a^3*b^2))/(2*a^5 *(b*(a - b))^(3/2)*(a^4 - 12*a^3*b - 96*a*b^3 + 48*b^4 + 60*a^2*b^2)) + (( (27*(a - 2*b)^3*(a*b - b^2)^(3/2)*(416*a^12*b - 16*a^13 + 768*a^10*b^3 - 1 152*a^11*b^2))/(64*a^20) - (3*(a - 2*b)*(a*b - b^2)^(1/2)*(27*a^8*b + 1728 *a^2*b^7 - 6048*a^3*b^6 + 8352*a^4*b^5 - 5760*a^5*b^4 + 2070*a^6*b^3 - 369 *a^7*b^2))/(4*a^12))*(4*a^4 - 60*a^3*b - 384*a*b^3 + 192*b^4 + 252*a^2*b^2 ))/(a^5*b*(144*a*b^4 - 13*a^4*b + a^5 - 48*b^5 - 156*a^2*b^3 + 72*a^3*b^2) )))/(27*a^2 - 108*a*b + 108*b^2) + (8*a^5*((27*(a - 2*b)^3*(a*b - b^2)^(3/ 2)*(32*a^14 - 128*a^13*b + 128*a^12*b^2))/(128*a^21) + (3*(a - 2*b)*(a*b - b^2)^(1/2)*(36*a^9*b - 1440*a^4*b^6 + 4320*a^5*b^5 - 4824*a^6*b^4 + 2448* a^7*b^3 - 540*a^8*b^2))/(8*a^13))*(4*a^4 - 60*a^3*b - 384*a*b^3 + 192*b...