Integrand size = 23, antiderivative size = 154 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {5 (3 a-7 b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{9/2} f}-\frac {(a-3 b) \cot (e+f x)}{a^4 f}-\frac {\cot ^3(e+f x)}{3 a^3 f}-\frac {(a-b) b \tan (e+f x)}{4 a^3 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {(7 a-11 b) b \tan (e+f x)}{8 a^4 f \left (a+b \tan ^2(e+f x)\right )} \]
-(a-3*b)*cot(f*x+e)/a^4/f-1/3*cot(f*x+e)^3/a^3/f-5/8*(3*a-7*b)*arctan(b^(1 /2)*tan(f*x+e)/a^(1/2))*b^(1/2)/a^(9/2)/f-1/4*(a-b)*b*tan(f*x+e)/a^3/f/(a+ b*tan(f*x+e)^2)^2-1/8*(7*a-11*b)*b*tan(f*x+e)/a^4/f/(a+b*tan(f*x+e)^2)
Time = 2.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {15 \sqrt {b} (-3 a+7 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )+\sqrt {a} \left (-8 \cot (e+f x) \left (2 a-9 b+a \csc ^2(e+f x)\right )-\frac {3 b \left (9 a^2-6 a b-11 b^2+\left (9 a^2-20 a b+11 b^2\right ) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{(a+b+(a-b) \cos (2 (e+f x)))^2}\right )}{24 a^{9/2} f} \]
(15*Sqrt[b]*(-3*a + 7*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]* (-8*Cot[e + f*x]*(2*a - 9*b + a*Csc[e + f*x]^2) - (3*b*(9*a^2 - 6*a*b - 11 *b^2 + (9*a^2 - 20*a*b + 11*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(24*a^(9/2)*f)
Time = 0.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4146, 361, 25, 1582, 1584, 2009}
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.
| \(\displaystyle \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^4 \left (a+b \tan (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\cot ^4(e+f x) \left (\tan ^2(e+f x)+1\right )}{\left (b \tan ^2(e+f x)+a\right )^3}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {-\frac {1}{4} b \int -\frac {\cot ^4(e+f x) \left (-\frac {3 (a-b) \tan ^4(e+f x)}{a^3}+\frac {4 (a-b) \tan ^2(e+f x)}{a^2 b}+\frac {4}{a b}\right )}{\left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)-\frac {b (a-b) \tan (e+f x)}{4 a^3 \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{4} b \int \frac {\cot ^4(e+f x) \left (-\frac {3 (a-b) \tan ^4(e+f x)}{a^3}+\frac {4 (a-b) \tan ^2(e+f x)}{a^2 b}+\frac {4}{a b}\right )}{\left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)-\frac {b (a-b) \tan (e+f x)}{4 a^3 \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \frac {\cot ^4(e+f x) \left (-\frac {(7 a-11 b) b^2 \tan ^4(e+f x)}{a}+8 (a-2 b) b \tan ^2(e+f x)+8 a b\right )}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{2 a^3 b^2}-\frac {(7 a-11 b) \tan (e+f x)}{2 a^4 \left (a+b \tan ^2(e+f x)\right )}\right )-\frac {b (a-b) \tan (e+f x)}{4 a^3 \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \left (8 b \cot ^4(e+f x)+\frac {8 (a-3 b) b \cot ^2(e+f x)}{a}-\frac {5 (3 a-7 b) b^2}{a \left (b \tan ^2(e+f x)+a\right )}\right )d\tan (e+f x)}{2 a^3 b^2}-\frac {(7 a-11 b) \tan (e+f x)}{2 a^4 \left (a+b \tan ^2(e+f x)\right )}\right )-\frac {b (a-b) \tan (e+f x)}{4 a^3 \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} b \left (\frac {-\frac {5 b^{3/2} (3 a-7 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {8 b (a-3 b) \cot (e+f x)}{a}-\frac {8}{3} b \cot ^3(e+f x)}{2 a^3 b^2}-\frac {(7 a-11 b) \tan (e+f x)}{2 a^4 \left (a+b \tan ^2(e+f x)\right )}\right )-\frac {b (a-b) \tan (e+f x)}{4 a^3 \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
(-1/4*((a - b)*b*Tan[e + f*x])/(a^3*(a + b*Tan[e + f*x]^2)^2) + (b*(((-5*( 3*a - 7*b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(3/2) - (8*(a - 3*b)*b*Cot[e + f*x])/a - (8*b*Cot[e + f*x]^3)/3)/(2*a^3*b^2) - ((7*a - 11*b)*Tan[e + f*x])/(2*a^4*(a + b*Tan[e + f*x]^2))))/4)/f
3.1.90.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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]
Time = 1.98 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a -3 b}{a^{4} \tan \left (f x +e \right )}-\frac {b \left (\frac {\left (\frac {7}{8} a b -\frac {11}{8} b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (9 a -13 b \right ) \tan \left (f x +e \right )}{8}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a -7 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(123\) |
| default | \(\frac {-\frac {1}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a -3 b}{a^{4} \tan \left (f x +e \right )}-\frac {b \left (\frac {\left (\frac {7}{8} a b -\frac {11}{8} b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (9 a -13 b \right ) \tan \left (f x +e \right )}{8}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a -7 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(123\) |
| risch | \(\frac {i \left (-16 a^{4}-105 b^{4} {\mathrm e}^{12 i \left (f x +e \right )}+48 a^{4} {\mathrm e}^{10 i \left (f x +e \right )}+630 b^{4} {\mathrm e}^{10 i \left (f x +e \right )}-105 b^{4}+325 a \,b^{3}+147 a^{3} b -351 a^{2} b^{2}-195 a^{2} b^{2} {\mathrm e}^{12 i \left (f x +e \right )}-260 a^{2} b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+502 \,{\mathrm e}^{2 i \left (f x +e \right )} b^{2} a^{2}-900 a \,b^{3} {\mathrm e}^{10 i \left (f x +e \right )}+255 a \,b^{3} {\mathrm e}^{12 i \left (f x +e \right )}+270 a^{2} b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+45 a^{3} b \,{\mathrm e}^{12 i \left (f x +e \right )}-135 a^{3} b \,{\mathrm e}^{8 i \left (f x +e \right )}+15 a^{2} b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+1375 a \,b^{3} {\mathrm e}^{8 i \left (f x +e \right )}-320 a^{3} b \,{\mathrm e}^{6 i \left (f x +e \right )}-1600 a \,b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-313 a^{3} b \,{\mathrm e}^{4 i \left (f x +e \right )}+19 a^{2} b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+1725 a \,b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+64 a^{3} b \,{\mathrm e}^{2 i \left (f x +e \right )}-1180 a \,b^{3} {\mathrm e}^{2 i \left (f x +e \right )}+176 a^{4} {\mathrm e}^{8 i \left (f x +e \right )}-1575 b^{4} {\mathrm e}^{8 i \left (f x +e \right )}+224 a^{4} {\mathrm e}^{6 i \left (f x +e \right )}+2100 b^{4} {\mathrm e}^{6 i \left (f x +e \right )}+96 a^{4} {\mathrm e}^{4 i \left (f x +e \right )}-1575 b^{4} {\mathrm e}^{4 i \left (f x +e \right )}-16 a^{4} {\mathrm e}^{2 i \left (f x +e \right )}+630 b^{4} {\mathrm e}^{2 i \left (f x +e \right )}\right )}{12 \left (a -b \right ) \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 )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} f \,a^{4}}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{16 a^{4} f}-\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{16 a^{5} f}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{16 a^{4} f}+\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{16 a^{5} f}\) | \(729\) |
1/f*(-1/3/a^3/tan(f*x+e)^3-(a-3*b)/a^4/tan(f*x+e)-1/a^4*b*(((7/8*a*b-11/8* b^2)*tan(f*x+e)^3+1/8*a*(9*a-13*b)*tan(f*x+e))/(a+b*tan(f*x+e)^2)^2+5/8*(3 *a-7*b)/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (138) = 276\).
Time = 0.36 (sec) , antiderivative size = 857, normalized size of antiderivative = 5.56 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\left [-\frac {4 \, {\left (16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 4 \, {\left (24 \, a^{3} - 206 \, a^{2} b + 485 \, a b^{2} - 315 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 20 \, {\left (15 \, a^{2} b - 62 \, a b^{2} + 63 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 3 \, a b^{2} + 7 \, b^{3} - {\left (6 \, a^{2} b - 23 \, a b^{2} + 21 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 60 \, {\left (3 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )}{96 \, {\left ({\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{6} - a^{4} b^{2} f - {\left (a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b - 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (24 \, a^{3} - 206 \, a^{2} b + 485 \, a b^{2} - 315 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 10 \, {\left (15 \, a^{2} b - 62 \, a b^{2} + 63 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 3 \, a b^{2} + 7 \, b^{3} - {\left (6 \, a^{2} b - 23 \, a b^{2} + 21 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 30 \, {\left (3 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )}{48 \, {\left ({\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{6} - a^{4} b^{2} f - {\left (a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b - 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ] \]
[-1/96*(4*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 4*(2 4*a^3 - 206*a^2*b + 485*a*b^2 - 315*b^3)*cos(f*x + e)^5 - 20*(15*a^2*b - 6 2*a*b^2 + 63*b^3)*cos(f*x + e)^3 + 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^ 3)*cos(f*x + e)^6 - (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^4 - 3*a*b^2 + 7*b^3 - (6*a^2*b - 23*a*b^2 + 21*b^3)*cos(f*x + e)^2)*sqrt(-b/ a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^ 2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e )^2 + b^2))*sin(f*x + e) - 60*(3*a*b^2 - 7*b^3)*cos(f*x + e))/(((a^6 - 2*a ^5*b + a^4*b^2)*f*cos(f*x + e)^6 - a^4*b^2*f - (a^6 - 4*a^5*b + 3*a^4*b^2) *f*cos(f*x + e)^4 - (2*a^5*b - 3*a^4*b^2)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/48*(2*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 2*(2 4*a^3 - 206*a^2*b + 485*a*b^2 - 315*b^3)*cos(f*x + e)^5 - 10*(15*a^2*b - 6 2*a*b^2 + 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^ 3)*cos(f*x + e)^6 - (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^4 - 3*a*b^2 + 7*b^3 - (6*a^2*b - 23*a*b^2 + 21*b^3)*cos(f*x + e)^2)*sqrt(b/a )*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f* x + e)))*sin(f*x + e) - 30*(3*a*b^2 - 7*b^3)*cos(f*x + e))/(((a^6 - 2*a^5* b + a^4*b^2)*f*cos(f*x + e)^6 - a^4*b^2*f - (a^6 - 4*a^5*b + 3*a^4*b^2)*f* cos(f*x + e)^4 - (2*a^5*b - 3*a^4*b^2)*f*cos(f*x + e)^2)*sin(f*x + e))]
Timed out. \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {15 \, {\left (3 \, a b^{2} - 7 \, b^{3}\right )} \tan \left (f x + e\right )^{6} + 25 \, {\left (3 \, a^{2} b - 7 \, a b^{2}\right )} \tan \left (f x + e\right )^{4} + 8 \, a^{3} + 8 \, {\left (3 \, a^{3} - 7 \, a^{2} b\right )} \tan \left (f x + e\right )^{2}}{a^{4} b^{2} \tan \left (f x + e\right )^{7} + 2 \, a^{5} b \tan \left (f x + e\right )^{5} + a^{6} \tan \left (f x + e\right )^{3}} + \frac {15 \, {\left (3 \, a b - 7 \, b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}}}{24 \, f} \]
-1/24*((15*(3*a*b^2 - 7*b^3)*tan(f*x + e)^6 + 25*(3*a^2*b - 7*a*b^2)*tan(f *x + e)^4 + 8*a^3 + 8*(3*a^3 - 7*a^2*b)*tan(f*x + e)^2)/(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3) + 15*(3*a*b - 7*b^2)* arctan(b*tan(f*x + e)/sqrt(a*b))/(sqrt(a*b)*a^4))/f
Time = 0.87 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {15 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (3 \, a b - 7 \, b^{2}\right )}}{\sqrt {a b} a^{4}} + \frac {3 \, {\left (7 \, a b^{2} \tan \left (f x + e\right )^{3} - 11 \, b^{3} \tan \left (f x + e\right )^{3} + 9 \, a^{2} b \tan \left (f x + e\right ) - 13 \, a b^{2} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2} a^{4}} + \frac {8 \, {\left (3 \, a \tan \left (f x + e\right )^{2} - 9 \, b \tan \left (f x + e\right )^{2} + a\right )}}{a^{4} \tan \left (f x + e\right )^{3}}}{24 \, f} \]
-1/24*(15*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqr t(a*b)))*(3*a*b - 7*b^2)/(sqrt(a*b)*a^4) + 3*(7*a*b^2*tan(f*x + e)^3 - 11* b^3*tan(f*x + e)^3 + 9*a^2*b*tan(f*x + e) - 13*a*b^2*tan(f*x + e))/((b*tan (f*x + e)^2 + a)^2*a^4) + 8*(3*a*tan(f*x + e)^2 - 9*b*tan(f*x + e)^2 + a)/ (a^4*tan(f*x + e)^3))/f
Time = 11.76 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (3\,a-7\,b\right )}{3\,a^2}+\frac {25\,b\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (3\,a-7\,b\right )}{24\,a^3}+\frac {5\,b^2\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (3\,a-7\,b\right )}{8\,a^4}}{f\,\left (a^2\,{\mathrm {tan}\left (e+f\,x\right )}^3+2\,a\,b\,{\mathrm {tan}\left (e+f\,x\right )}^5+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^7\right )}-\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a}}\right )\,\left (3\,a-7\,b\right )}{8\,a^{9/2}\,f} \]