Integrand size = 23, antiderivative size = 205 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\sqrt {b} \left (15 a^2-40 a b+24 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^4 (a-b)^{3/2} f}-\frac {(a-6 b) \text {arctanh}(\cos (e+f x))}{2 a^4 f}-\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 b \sec (e+f x)}{4 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(11 a-12 b) b \sec (e+f x)}{8 a^3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )} \]
-1/2*(a-6*b)*arctanh(cos(f*x+e))/a^4/f-1/2*cot(f*x+e)*csc(f*x+e)/a/f/(a-b+ b*sec(f*x+e)^2)^2-3/4*b*sec(f*x+e)/a^2/f/(a-b+b*sec(f*x+e)^2)^2-1/8*(11*a- 12*b)*b*sec(f*x+e)/a^3/(a-b)/f/(a-b+b*sec(f*x+e)^2)-1/8*(15*a^2-40*a*b+24* b^2)*arctan(sec(f*x+e)*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^4/(a-b)^(3/2)/f
Leaf count is larger than twice the leaf count of optimal. \(414\) vs. \(2(205)=410\).
Time = 7.65 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.02 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {\sqrt {a-b} \sqrt {b} \left (15 a^2-40 a b+24 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^4 (-a+b)^2 f}+\frac {\sqrt {a-b} \sqrt {b} \left (15 a^2-40 a b+24 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^4 (-a+b)^2 f}+\frac {b^2 \cos (e+f x)}{a^2 (a-b) f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}+\frac {-9 a b \cos (e+f x)+8 b^2 \cos (e+f x)}{4 a^3 (a-b) f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}-\frac {\csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 a^3 f}+\frac {(-a+6 b) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 a^4 f}+\frac {(a-6 b) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 a^4 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 a^3 f} \]
(Sqrt[a - b]*Sqrt[b]*(15*a^2 - 40*a*b + 24*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^4 *(-a + b)^2*f) + (Sqrt[a - b]*Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(S ec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2])) /Sqrt[b]])/(8*a^4*(-a + b)^2*f) + (b^2*Cos[e + f*x])/(a^2*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) + (-9*a*b*Cos[e + f*x] + 8* b^2*Cos[e + f*x])/(4*a^3*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*( e + f*x)])) - Csc[(e + f*x)/2]^2/(8*a^3*f) + ((-a + 6*b)*Log[Cos[(e + f*x) /2]])/(2*a^4*f) + ((a - 6*b)*Log[Sin[(e + f*x)/2]])/(2*a^4*f) + Sec[(e + f *x)/2]^2/(8*a^3*f)
Time = 0.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4147, 373, 402, 27, 402, 25, 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.
| \(\displaystyle \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^3 \left (a+b \tan (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int \frac {\sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\int \frac {-5 b \sec ^2(e+f x)+a-b}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^3}d\sec (e+f x)}{2 a}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\int -\frac {2 (a-b) \left (-9 b \sec ^2(e+f x)+2 a-3 b\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{4 a (a-b)}}{2 a}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\int \frac {-9 b \sec ^2(e+f x)+2 a-3 b}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )^2}d\sec (e+f x)}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\frac {b (11 a-12 b) \sec (e+f x)}{2 a (a-b) \left (a+b \sec ^2(e+f x)-b\right )}-\frac {\int -\frac {4 a^2-17 b a+12 b^2-(11 a-12 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{2 a (a-b)}}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\frac {\int \frac {4 a^2-17 b a+12 b^2-(11 a-12 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a-b\right )}d\sec (e+f x)}{2 a (a-b)}+\frac {b (11 a-12 b) \sec (e+f x)}{2 a (a-b) \left (a+b \sec ^2(e+f x)-b\right )}}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\frac {\frac {b \left (15 a^2-40 a b+24 b^2\right ) \int \frac {1}{b \sec ^2(e+f x)+a-b}d\sec (e+f x)}{a}+\frac {4 (a-6 b) (a-b) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}}{2 a (a-b)}+\frac {b (11 a-12 b) \sec (e+f x)}{2 a (a-b) \left (a+b \sec ^2(e+f x)-b\right )}}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\frac {\frac {4 (a-6 b) (a-b) \int \frac {1}{1-\sec ^2(e+f x)}d\sec (e+f x)}{a}+\frac {\sqrt {b} \left (15 a^2-40 a b+24 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a (a-b)}+\frac {b (11 a-12 b) \sec (e+f x)}{2 a (a-b) \left (a+b \sec ^2(e+f x)-b\right )}}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{2 a \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {\frac {\frac {\frac {\sqrt {b} \left (15 a^2-40 a b+24 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {4 (a-6 b) (a-b) \text {arctanh}(\sec (e+f x))}{a}}{2 a (a-b)}+\frac {b (11 a-12 b) \sec (e+f x)}{2 a (a-b) \left (a+b \sec ^2(e+f x)-b\right )}}{2 a}+\frac {3 b \sec (e+f x)}{2 a \left (a+b \sec ^2(e+f x)-b\right )^2}}{2 a}}{f}\) |
(Sec[e + f*x]/(2*a*(1 - Sec[e + f*x]^2)*(a - b + b*Sec[e + f*x]^2)^2) - (( 3*b*Sec[e + f*x])/(2*a*(a - b + b*Sec[e + f*x]^2)^2) + (((Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a*Sqrt[a - b]) + (4*(a - 6*b)*(a - b)*ArcTanh[Sec[e + f*x]])/a)/(2*a*(a - b)) + ((11 *a - 12*b)*b*Sec[e + f*x])/(2*a*(a - b)*(a - b + b*Sec[e + f*x]^2)))/(2*a) )/(2*a))/f
3.1.84.3.1 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[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[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 = 1.67 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {-\frac {\left (9 a -8 b \right ) a \cos \left (f x +e \right )^{3}}{8}-\frac {a b \left (7 a -8 b \right ) \cos \left (f x +e \right )}{8 \left (a -b \right )}}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {\left (15 a^{2}-40 a b +24 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \left (a -b \right ) \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{4 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a +6 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{4 a^{4}}+\frac {1}{4 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (a -6 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{4 a^{4}}}{f}\) | \(206\) |
| default | \(\frac {\frac {b \left (\frac {-\frac {\left (9 a -8 b \right ) a \cos \left (f x +e \right )^{3}}{8}-\frac {a b \left (7 a -8 b \right ) \cos \left (f x +e \right )}{8 \left (a -b \right )}}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {\left (15 a^{2}-40 a b +24 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \left (a -b \right ) \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{4 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a +6 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{4 a^{4}}+\frac {1}{4 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (a -6 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{4 a^{4}}}{f}\) | \(206\) |
| risch | \(\text {Expression too large to display}\) | \(913\) |
1/f*(b/a^4*((-1/8*(9*a-8*b)*a*cos(f*x+e)^3-1/8*a*b*(7*a-8*b)/(a-b)*cos(f*x +e))/(a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)^2+1/8*(15*a^2-40*a*b+24*b^2)/(a-b)/ (b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2)))+1/4/a^3/(cos(f*x +e)+1)+1/4/a^4*(-a+6*b)*ln(cos(f*x+e)+1)+1/4/a^3/(cos(f*x+e)-1)+1/4*(a-6*b )/a^4*ln(cos(f*x+e)-1))
Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (187) = 374\).
Time = 0.48 (sec) , antiderivative size = 1419, normalized size of antiderivative = 6.92 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
[1/16*(2*(4*a^4 - 21*a^3*b + 29*a^2*b^2 - 12*a*b^3)*cos(f*x + e)^5 + 2*(17 *a^3*b - 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^3 - ((15*a^4 - 70*a^3*b + 119 *a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*a^3*b + 229*a ^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f*x + e)^4 - 15*a^2*b^2 + 40*a*b^3 - 24*b ^4 - (30*a^3*b - 125*a^2*b^2 + 168*a*b^3 - 72*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)*cos(f*x + e)^2 + b)) + 2*(11*a^2*b^2 - 12*a*b^3)*cos( f*x + e) - 4*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e) ^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^ 2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f *x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 4*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2* b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a ^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7* a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)* f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f), 1/8*((4*a^4 - 21*a^3*b + 29*a^2 *b^2 - 12*a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 40*a^2*b^2 + 24*a*b^3)*cos(f *x + e)^3 - ((15*a^4 - 70*a^3*b + 119*a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*a^3*b + 229*a^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f...
Timed out. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 583 vs. \(2 (187) = 374\).
Time = 0.86 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.84 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {{\left (15 \, a^{2} b - 40 \, a b^{2} + 24 \, 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 )}{{\left (a^{5} - a^{4} b\right )} \sqrt {a b - b^{2}}} + \frac {2 \, {\left (9 \, a^{3} b - 10 \, a^{2} b^{2} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {80 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {56 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {102 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {152 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {80 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {32 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {24 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{{\left (a^{5} - a^{4} b\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 )}^{2}} - \frac {2 \, {\left (a - 6 \, b\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 {{\left (a - \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {12 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}} + \frac {\cos \left (f x + e\right ) - 1}{a^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}}{8 \, f} \]
-1/8*((15*a^2*b - 40*a*b^2 + 24*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)))/((a^5 - a^4*b)* sqrt(a*b - b^2)) + 2*(9*a^3*b - 10*a^2*b^2 + 27*a^3*b*(cos(f*x + e) - 1)/( cos(f*x + e) + 1) - 80*a^2*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 56* a*b^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 27*a^3*b*(cos(f*x + e) - 1)^ 2/(cos(f*x + e) + 1)^2 - 102*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 152*a*b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 80*b^4*(cos(f *x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 9*a^3*b*(cos(f*x + e) - 1)^3/(cos(f* x + e) + 1)^3 - 32*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 24* a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)/((a^5 - a^4*b)*(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)^2) - 2*(a - 6*b)*log (abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/a^4 - (a - 2*a*(cos(f*x + e ) - 1)/(cos(f*x + e) + 1) + 12*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1))*(c os(f*x + e) + 1)/(a^4*(cos(f*x + e) - 1)) + (cos(f*x + e) - 1)/(a^3*(cos(f *x + e) + 1)))/f
Time = 12.70 (sec) , antiderivative size = 1652, normalized size of antiderivative = 8.06 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
tan(e/2 + (f*x)/2)^2/(8*a^3*f) - (a^2/2 + (tan(e/2 + (f*x)/2)^4*(96*a*b^2 - 38*a^2*b + 3*a^3 - 64*b^3))/(a - b) + (tan(e/2 + (f*x)/2)^8*(64*a*b^2 - 19*a^2*b + a^3 - 48*b^3))/(2*(a - b)) - (tan(e/2 + (f*x)/2)^2*(14*a*b^2 - 15*a^2*b + 2*a^3))/(a - b) - (tan(e/2 + (f*x)/2)^6*(2*a^4 - 33*a^3*b - 152 *a*b^3 + 80*b^4 + 106*a^2*b^2))/(a*(a - b)))/(f*(4*a^5*tan(e/2 + (f*x)/2)^ 2 + 4*a^5*tan(e/2 + (f*x)/2)^10 + tan(e/2 + (f*x)/2)^6*(24*a^5 - 64*a^4*b + 64*a^3*b^2) + tan(e/2 + (f*x)/2)^4*(32*a^4*b - 16*a^5) + tan(e/2 + (f*x) /2)^8*(32*a^4*b - 16*a^5))) + (log(tan(e/2 + (f*x)/2))*(a - 6*b))/(2*a^4*f ) + (b^(1/2)*atan(((tan(e/2 + (f*x)/2)^2*((((b^(3/2)*(15*a^2 - 40*a*b + 24 *b^2)^3*(128*a^16 - 3712*a^15*b + 6144*a^10*b^6 - 27648*a^11*b^5 + 49408*a ^12*b^4 - 43904*a^13*b^3 + 19584*a^14*b^2))/(32768*a^12*(a - b)^(9/2)*(3*a ^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)) + (b^(1/2)*(15*a^2 - 40*a*b + 24*b^2) *(360*a^9*b - 13824*a^2*b^8 + 66816*a^3*b^7 - 132864*a^4*b^6 + 139776*a^5* b^5 - 83240*a^6*b^4 + 27836*a^7*b^3 - 4860*a^8*b^2))/(128*a^4*(a - b)^(3/2 )*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))*(63*a^6 - 1013*a^5*b - 9600*a* b^5 + 2304*b^6 + 15792*a^2*b^4 - 12888*a^3*b^3 + 5342*a^4*b^2))/(2*a^5*(a - b)^(9/2)*(5760*a*b^4 - 735*a^4*b + 64*a^5 - 1728*b^5 - 6960*a^2*b^3 + 36 00*a^3*b^2)) - (((6912*a*b^6 - 1728*b^7 - 10800*a^2*b^5 + 8240*a^3*b^4 - 3 075*a^4*b^3 + 450*a^5*b^2)/(8*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)) + ( b*(15*a^2 - 40*a*b + 24*b^2)^2*(1936*a^12*b - 64*a^13 + 18432*a^6*b^7 -...