Integrand size = 23, antiderivative size = 121 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (3 a^2+30 a b+35 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {(a+b) (3 a+11 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {(a+b)^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac {b (2 a+3 b) \sec (e+f x)}{f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \] Output:
-1/8*(3*a^2+30*a*b+35*b^2)*arctanh(cos(f*x+e))/f-1/8*(a+b)*(3*a+11*b)*cot( f*x+e)*csc(f*x+e)/f-1/4*(a+b)^2*cot(f*x+e)*csc(f*x+e)^3/f+b*(2*a+3*b)*sec( f*x+e)/f+1/3*b^2*sec(f*x+e)^3/f
Time = 1.96 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.80 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (b+a \cos ^2(e+f x)\right )^2 \left (\left (90 a^2+132 a b-102 b^2+\left (6 a^2+60 a b+70 b^2\right ) \cos (4 (e+f x))-3 \left (3 a^2+30 a b+35 b^2\right ) \cos (6 (e+f x))\right ) \cot (e+f x) \csc ^3(e+f x)+\frac {1}{2} \left (105 a^2+282 a b+329 b^2\right ) (\cos (e+f x)+\cos (3 (e+f x))) \csc ^4(e+f x)+96 \left (3 a^2+30 a b+35 b^2\right ) \cos ^4(e+f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) \sec ^4(e+f x)}{192 f (a+2 b+a \cos (2 (e+f x)))^2} \] Input:
Integrate[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]
Output:
-1/192*((b + a*Cos[e + f*x]^2)^2*((90*a^2 + 132*a*b - 102*b^2 + (6*a^2 + 6 0*a*b + 70*b^2)*Cos[4*(e + f*x)] - 3*(3*a^2 + 30*a*b + 35*b^2)*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^3 + ((105*a^2 + 282*a*b + 329*b^2)*(Cos[e + f*x] + Cos[3*(e + f*x)])*Csc[e + f*x]^4)/2 + 96*(3*a^2 + 30*a*b + 35*b^ 2)*Cos[e + f*x]^4*(Log[Cos[(e + f*x)/2]] - Log[Sin[(e + f*x)/2]]))*Sec[e + f*x]^4)/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4621, 365, 361, 25, 361, 25, 359, 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.
| \(\displaystyle \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sec (e+f x)^2\right )^2}{\sin (e+f x)^5}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\left (a \cos ^2(e+f x)+b\right )^2 \sec ^4(e+f x)}{\left (1-\cos ^2(e+f x)\right )^3}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle -\frac {\frac {1}{3} \int \frac {\left (3 a^2 \cos ^2(e+f x)+b (6 a+7 b)\right ) \sec ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right )^3}d\cos (e+f x)-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}-\frac {1}{4} \int -\frac {\left (3 \left (3 a^2+6 b a+7 b^2\right ) \cos ^2(e+f x)+4 b (6 a+7 b)\right ) \sec ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right )^2}d\cos (e+f x)\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {1}{4} \int \frac {\left (3 \left (3 a^2+6 b a+7 b^2\right ) \cos ^2(e+f x)+4 b (6 a+7 b)\right ) \sec ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right )^2}d\cos (e+f x)+\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {1}{4} \left (\frac {(3 a+7 b)^2 \cos (e+f x)}{2 \left (1-\cos ^2(e+f x)\right )}-\frac {1}{2} \int -\frac {\left ((3 a+7 b)^2 \cos ^2(e+f x)+8 b (6 a+7 b)\right ) \sec ^2(e+f x)}{1-\cos ^2(e+f x)}d\cos (e+f x)\right )+\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {\left ((3 a+7 b)^2 \cos ^2(e+f x)+8 b (6 a+7 b)\right ) \sec ^2(e+f x)}{1-\cos ^2(e+f x)}d\cos (e+f x)+\frac {(3 a+7 b)^2 \cos (e+f x)}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 \left (3 a^2+30 a b+35 b^2\right ) \int \frac {1}{1-\cos ^2(e+f x)}d\cos (e+f x)-8 b (6 a+7 b) \sec (e+f x)\right )+\frac {(3 a+7 b)^2 \cos (e+f x)}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 \left (3 a^2+30 a b+35 b^2\right ) \text {arctanh}(\cos (e+f x))-8 b (6 a+7 b) \sec (e+f x)\right )+\frac {(3 a+7 b)^2 \cos (e+f x)}{2 \left (1-\cos ^2(e+f x)\right )}\right )+\frac {\left (3 a^2+6 a b+7 b^2\right ) \cos (e+f x)}{4 \left (1-\cos ^2(e+f x)\right )^2}\right )-\frac {b^2 \sec ^3(e+f x)}{3 \left (1-\cos ^2(e+f x)\right )^2}}{f}\) |
Input:
Int[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]
Output:
-((-1/3*(b^2*Sec[e + f*x]^3)/(1 - Cos[e + f*x]^2)^2 + (((3*a^2 + 6*a*b + 7 *b^2)*Cos[e + f*x])/(4*(1 - Cos[e + f*x]^2)^2) + (((3*a + 7*b)^2*Cos[e + f *x])/(2*(1 - Cos[e + f*x]^2)) + (3*(3*a^2 + 30*a*b + 35*b^2)*ArcTanh[Cos[e + f*x]] - 8*b*(6*a + 7*b)*Sec[e + f*x])/2)/4)/3)/f)
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), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 1.50 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.74
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {1}{4 \sin \left (f x +e \right )^{4} \cos \left (f x +e \right )}-\frac {5}{8 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {15}{8 \cos \left (f x +e \right )}+\frac {15 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\frac {1}{4 \sin \left (f x +e \right )^{4} \cos \left (f x +e \right )^{3}}+\frac {7}{12 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{3}}-\frac {35}{24 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {35}{8 \cos \left (f x +e \right )}+\frac {35 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )}{f}\) | \(211\) |
| default | \(\frac {a^{2} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {1}{4 \sin \left (f x +e \right )^{4} \cos \left (f x +e \right )}-\frac {5}{8 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {15}{8 \cos \left (f x +e \right )}+\frac {15 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\frac {1}{4 \sin \left (f x +e \right )^{4} \cos \left (f x +e \right )^{3}}+\frac {7}{12 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{3}}-\frac {35}{24 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {35}{8 \cos \left (f x +e \right )}+\frac {35 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )}{f}\) | \(211\) |
| norman | \(\frac {\frac {a^{2}+2 a b +b^{2}}{64 f}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{64 f}+\frac {\left (5 a^{2}+26 a b +21 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{64 f}+\frac {\left (5 a^{2}+26 a b +21 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{64 f}-\frac {\left (3 a^{2}+29 a b +42 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{4 f}+\frac {\left (39 a^{2}+422 a b +511 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{32 f}-\frac {\left (63 a^{2}+654 a b +847 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{96 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {\left (3 a^{2}+30 a b +35 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}\) | \(258\) |
| parallelrisch | \(\frac {55296 \left (\frac {\cos \left (3 f x +3 e \right )}{3}+\cos \left (f x +e \right )\right ) \left (a^{2}+10 a b +\frac {35}{3} b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-63 \left (\left (3 a^{2}+\frac {218}{7} a b +\frac {121}{3} b^{2}\right ) \cos \left (3 f x +3 e \right )+\left (\frac {80}{3} a^{2}+\frac {1504}{21} a b +\frac {752}{9} b^{2}\right ) \cos \left (2 f x +2 e \right )+\left (\frac {32}{21} a^{2}+\frac {320}{21} a b +\frac {160}{9} b^{2}\right ) \cos \left (4 f x +4 e \right )+\left (a^{2}+\frac {218}{21} a b +\frac {121}{9} b^{2}\right ) \cos \left (5 f x +5 e \right )+\left (-\frac {16}{7} a^{2}-\frac {160}{7} a b -\frac {80}{3} b^{2}\right ) \cos \left (6 f x +6 e \right )+\left (-a^{2}-\frac {218}{21} a b -\frac {121}{9} b^{2}\right ) \cos \left (7 f x +7 e \right )+\left (-3 a^{2}-\frac {218}{7} a b -\frac {121}{3} b^{2}\right ) \cos \left (f x +e \right )+\frac {160 a^{2}}{7}+\frac {704 a b}{21}-\frac {544 b^{2}}{21}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{49152 f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(279\) |
| risch | \(\frac {{\mathrm e}^{i \left (f x +e \right )} \left (9 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}+90 a b \,{\mathrm e}^{12 i \left (f x +e \right )}+105 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}-6 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}-60 a b \,{\mathrm e}^{10 i \left (f x +e \right )}-70 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}-105 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}-282 a b \,{\mathrm e}^{8 i \left (f x +e \right )}-329 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-180 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}-264 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+204 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-105 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-282 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-329 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-6 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-60 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-70 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{2}+90 a b +105 b^{2}\right )}{12 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) a^{2}}{8 f}+\frac {15 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) a b}{4 f}+\frac {35 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{8 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) a^{2}}{8 f}-\frac {15 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) a b}{4 f}-\frac {35 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{8 f}\) | \(461\) |
Input:
int(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/f*(a^2*((-1/4*csc(f*x+e)^3-3/8*csc(f*x+e))*cot(f*x+e)+3/8*ln(csc(f*x+e)- cot(f*x+e)))+2*a*b*(-1/4/sin(f*x+e)^4/cos(f*x+e)-5/8/sin(f*x+e)^2/cos(f*x+ e)+15/8/cos(f*x+e)+15/8*ln(csc(f*x+e)-cot(f*x+e)))+b^2*(-1/4/sin(f*x+e)^4/ cos(f*x+e)^3+7/12/sin(f*x+e)^2/cos(f*x+e)^3-35/24/sin(f*x+e)^2/cos(f*x+e)+ 35/8/cos(f*x+e)+35/8*ln(csc(f*x+e)-cot(f*x+e))))
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (113) = 226\).
Time = 0.09 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.36 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {6 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 10 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (6 \, a b + 7 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 16 \, b^{2} - 3 \, {\left ({\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{48 \, {\left (f \cos \left (f x + e\right )^{7} - 2 \, f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{3}\right )}} \] Input:
integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")
Output:
1/48*(6*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^6 - 10*(3*a^2 + 30*a*b + 35 *b^2)*cos(f*x + e)^4 + 16*(6*a*b + 7*b^2)*cos(f*x + e)^2 + 16*b^2 - 3*((3* a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 30*a*b + 35*b^2)*cos(f* x + e)^5 + (3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^5 + (3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^3)*log(-1 /2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)
Timed out. \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)**5*(a+b*sec(f*x+e)**2)**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.36 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {3 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - 3 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (6 \, a b + 7 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )}}{\cos \left (f x + e\right )^{7} - 2 \, \cos \left (f x + e\right )^{5} + \cos \left (f x + e\right )^{3}}}{48 \, f} \] Input:
integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")
Output:
-1/48*(3*(3*a^2 + 30*a*b + 35*b^2)*log(cos(f*x + e) + 1) - 3*(3*a^2 + 30*a *b + 35*b^2)*log(cos(f*x + e) - 1) - 2*(3*(3*a^2 + 30*a*b + 35*b^2)*cos(f* x + e)^6 - 5*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^4 + 8*(6*a*b + 7*b^2)* cos(f*x + e)^2 + 8*b^2)/(cos(f*x + e)^7 - 2*cos(f*x + e)^5 + cos(f*x + e)^ 3))/f
Time = 0.18 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.58 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {{\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \cos \left (f x + e\right ) + 1 \right |}\right )}{16 \, f} + \frac {{\left (3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \cos \left (f x + e\right ) - 1 \right |}\right )}{16 \, f} + \frac {3 \, a^{2} \cos \left (f x + e\right )^{3} + 14 \, a b \cos \left (f x + e\right )^{3} + 11 \, b^{2} \cos \left (f x + e\right )^{3} - 5 \, a^{2} \cos \left (f x + e\right ) - 18 \, a b \cos \left (f x + e\right ) - 13 \, b^{2} \cos \left (f x + e\right )}{8 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )}^{2} f} + \frac {6 \, a b \cos \left (f x + e\right )^{2} + 9 \, b^{2} \cos \left (f x + e\right )^{2} + b^{2}}{3 \, f \cos \left (f x + e\right )^{3}} \] Input:
integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Output:
-1/16*(3*a^2 + 30*a*b + 35*b^2)*log(abs(cos(f*x + e) + 1))/f + 1/16*(3*a^2 + 30*a*b + 35*b^2)*log(abs(cos(f*x + e) - 1))/f + 1/8*(3*a^2*cos(f*x + e) ^3 + 14*a*b*cos(f*x + e)^3 + 11*b^2*cos(f*x + e)^3 - 5*a^2*cos(f*x + e) - 18*a*b*cos(f*x + e) - 13*b^2*cos(f*x + e))/((cos(f*x + e)^2 - 1)^2*f) + 1/ 3*(6*a*b*cos(f*x + e)^2 + 9*b^2*cos(f*x + e)^2 + b^2)/(f*cos(f*x + e)^3)
Time = 12.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\frac {b^2}{3}+{\cos \left (e+f\,x\right )}^2\,\left (\frac {7\,b^2}{3}+2\,a\,b\right )+{\cos \left (e+f\,x\right )}^6\,\left (\frac {3\,a^2}{8}+\frac {15\,a\,b}{4}+\frac {35\,b^2}{8}\right )-{\cos \left (e+f\,x\right )}^4\,\left (\frac {5\,a^2}{8}+\frac {25\,a\,b}{4}+\frac {175\,b^2}{24}\right )}{f\,\left ({\cos \left (e+f\,x\right )}^7-2\,{\cos \left (e+f\,x\right )}^5+{\cos \left (e+f\,x\right )}^3\right )}-\frac {\mathrm {atanh}\left (\cos \left (e+f\,x\right )\right )\,\left (\frac {3\,a^2}{8}+\frac {15\,a\,b}{4}+\frac {35\,b^2}{8}\right )}{f} \] Input:
int((a + b/cos(e + f*x)^2)^2/sin(e + f*x)^5,x)
Output:
(b^2/3 + cos(e + f*x)^2*(2*a*b + (7*b^2)/3) + cos(e + f*x)^6*((15*a*b)/4 + (3*a^2)/8 + (35*b^2)/8) - cos(e + f*x)^4*((25*a*b)/4 + (5*a^2)/8 + (175*b ^2)/24))/(f*(cos(e + f*x)^3 - 2*cos(e + f*x)^5 + cos(e + f*x)^7)) - (atanh (cos(e + f*x))*((15*a*b)/4 + (3*a^2)/8 + (35*b^2)/8))/f
Time = 0.16 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.69 \[ \int \csc ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {72 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{6} a^{2}+720 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{6} a b +840 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{6} b^{2}-72 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{4} a^{2}-720 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{4} a b -840 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right )^{4} b^{2}-63 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a^{2}-654 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a b -847 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b^{2}+63 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a^{2}+654 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a b +847 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b^{2}+72 \sin \left (f x +e \right )^{6} a^{2}+720 \sin \left (f x +e \right )^{6} a b +840 \sin \left (f x +e \right )^{6} b^{2}-96 \sin \left (f x +e \right )^{4} a^{2}-960 \sin \left (f x +e \right )^{4} a b -1120 \sin \left (f x +e \right )^{4} b^{2}-24 \sin \left (f x +e \right )^{2} a^{2}+144 \sin \left (f x +e \right )^{2} a b +168 \sin \left (f x +e \right )^{2} b^{2}+48 a^{2}+96 a b +48 b^{2}}{192 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} f \left (\sin \left (f x +e \right )^{2}-1\right )} \] Input:
int(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x)
Output:
(72*cos(e + f*x)*log(tan((e + f*x)/2))*sin(e + f*x)**6*a**2 + 720*cos(e + f*x)*log(tan((e + f*x)/2))*sin(e + f*x)**6*a*b + 840*cos(e + f*x)*log(tan( (e + f*x)/2))*sin(e + f*x)**6*b**2 - 72*cos(e + f*x)*log(tan((e + f*x)/2)) *sin(e + f*x)**4*a**2 - 720*cos(e + f*x)*log(tan((e + f*x)/2))*sin(e + f*x )**4*a*b - 840*cos(e + f*x)*log(tan((e + f*x)/2))*sin(e + f*x)**4*b**2 - 6 3*cos(e + f*x)*sin(e + f*x)**6*a**2 - 654*cos(e + f*x)*sin(e + f*x)**6*a*b - 847*cos(e + f*x)*sin(e + f*x)**6*b**2 + 63*cos(e + f*x)*sin(e + f*x)**4 *a**2 + 654*cos(e + f*x)*sin(e + f*x)**4*a*b + 847*cos(e + f*x)*sin(e + f* x)**4*b**2 + 72*sin(e + f*x)**6*a**2 + 720*sin(e + f*x)**6*a*b + 840*sin(e + f*x)**6*b**2 - 96*sin(e + f*x)**4*a**2 - 960*sin(e + f*x)**4*a*b - 1120 *sin(e + f*x)**4*b**2 - 24*sin(e + f*x)**2*a**2 + 144*sin(e + f*x)**2*a*b + 168*sin(e + f*x)**2*b**2 + 48*a**2 + 96*a*b + 48*b**2)/(192*cos(e + f*x) *sin(e + f*x)**4*f*(sin(e + f*x)**2 - 1))