Integrand size = 23, antiderivative size = 257 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 \sqrt {a} (a+b)^5 f}-\frac {3 \left (a^2-10 a b+5 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 (a+b)^5 f}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{8 (a+b)^3 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {3 \left (a^2-6 a b+b^2\right ) \cos (e+f x)}{8 (a+b)^4 f \left (b+a \cos ^2(e+f x)\right )}-\frac {(a-7 b) \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2} \] Output:
3/8*b^(1/2)*(5*a^2-10*a*b+b^2)*arctan(a^(1/2)*cos(f*x+e)/b^(1/2))/a^(1/2)/ (a+b)^5/f-3/8*(a^2-10*a*b+5*b^2)*arctanh(cos(f*x+e))/(a+b)^5/f+1/8*(a^2-9* a*b+2*b^2)*cos(f*x+e)/(a+b)^3/f/(b+a*cos(f*x+e)^2)^2+3/8*(a^2-6*a*b+b^2)*c os(f*x+e)/(a+b)^4/f/(b+a*cos(f*x+e)^2)-1/8*(a-7*b)*cot(f*x+e)*csc(f*x+e)/( a+b)^2/f/(b+a*cos(f*x+e)^2)^2-1/4*cot(f*x+e)^3*csc(f*x+e)/(a+b)/f/(b+a*cos (f*x+e)^2)^2
Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.14 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \left (\frac {48 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2}{\sqrt {a}}+\frac {48 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2}{\sqrt {a}}-2 (a+b) \left (30 a^3+112 a^2 b+182 a b^2-140 b^3+\left (35 a^3+78 a^2 b-93 a b^2+224 b^3\right ) \cos (2 (e+f x))+2 \left (a^3-8 a^2 b+53 a b^2-10 b^3\right ) \cos (4 (e+f x))-3 a^3 \cos (6 (e+f x))+18 a^2 b \cos (6 (e+f x))-3 a b^2 \cos (6 (e+f x))\right ) \cot (e+f x) \csc ^3(e+f x)-48 \left (a^2-10 a b+5 b^2\right ) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+48 \left (a^2-10 a b+5 b^2\right ) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^6(e+f x)}{1024 (a+b)^5 f \left (a+b \sec ^2(e+f x)\right )^3} \] Input:
Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]
Output:
((a + 2*b + a*Cos[2*(e + f*x)])*((48*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan [((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/ 2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2 ]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2)/Sqrt[a] + (48*Sqrt[b]*(5*a^ 2 - 10*a*b + b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e ])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2)/Sqr t[a] - 2*(a + b)*(30*a^3 + 112*a^2*b + 182*a*b^2 - 140*b^3 + (35*a^3 + 78* a^2*b - 93*a*b^2 + 224*b^3)*Cos[2*(e + f*x)] + 2*(a^3 - 8*a^2*b + 53*a*b^2 - 10*b^3)*Cos[4*(e + f*x)] - 3*a^3*Cos[6*(e + f*x)] + 18*a^2*b*Cos[6*(e + f*x)] - 3*a*b^2*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^3 - 48*(a^2 - 10*a*b + 5*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Cos[(e + f*x)/2]] + 48*(a^2 - 10*a*b + 5*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Sin[(e + f* x)/2]])*Sec[e + f*x]^6)/(1024*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)
Time = 0.52 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4621, 372, 440, 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.
| \(\displaystyle \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^5 \left (a+b \sec (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\cos ^6(e+f x)}{\left (1-\cos ^2(e+f x)\right )^3 \left (a \cos ^2(e+f x)+b\right )^3}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\int \frac {\cos ^2(e+f x) \left (3 b-(a-4 b) \cos ^2(e+f x)\right )}{\left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^3}d\cos (e+f x)}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\int \frac {(a-7 b) b-\left (3 a^2-29 b a+8 b^2\right ) \cos ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^3}d\cos (e+f x)}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\int -\frac {12 b \left ((a-3 b) b-\left (a^2-9 b a+2 b^2\right ) \cos ^2(e+f x)\right )}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}d\cos (e+f x)}{4 b (a+b)}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \int \frac {(a-3 b) b-\left (a^2-9 b a+2 b^2\right ) \cos ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}d\cos (e+f x)}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \left (\frac {\left (a^2-6 a b+b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )}-\frac {\int -\frac {2 b \left (4 (a-b) b-\left (a^2-6 b a+b^2\right ) \cos ^2(e+f x)\right )}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )}d\cos (e+f x)}{2 b (a+b)}\right )}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \left (\frac {\int \frac {4 (a-b) b-\left (a^2-6 b a+b^2\right ) \cos ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )}d\cos (e+f x)}{a+b}+\frac {\left (a^2-6 a b+b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )}\right )}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \left (\frac {\frac {b \left (5 a^2-10 a b+b^2\right ) \int \frac {1}{a \cos ^2(e+f x)+b}d\cos (e+f x)}{a+b}-\frac {\left (a^2-10 a b+5 b^2\right ) \int \frac {1}{1-\cos ^2(e+f x)}d\cos (e+f x)}{a+b}}{a+b}+\frac {\left (a^2-6 a b+b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )}\right )}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \left (\frac {\frac {\sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {\left (a^2-10 a b+5 b^2\right ) \int \frac {1}{1-\cos ^2(e+f x)}d\cos (e+f x)}{a+b}}{a+b}+\frac {\left (a^2-6 a b+b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )}\right )}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\cos ^3(e+f x)}{4 (a+b) \left (1-\cos ^2(e+f x)\right )^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {3 \left (\frac {\frac {\sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {\left (a^2-10 a b+5 b^2\right ) \text {arctanh}(\cos (e+f x))}{a+b}}{a+b}+\frac {\left (a^2-6 a b+b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )}\right )}{a+b}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{(a+b) \left (a \cos ^2(e+f x)+b\right )^2}}{2 (a+b)}-\frac {(a-7 b) \cos (e+f x)}{2 (a+b) \left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}}{4 (a+b)}}{f}\) |
Input:
Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]
Output:
-((Cos[e + f*x]^3/(4*(a + b)*(1 - Cos[e + f*x]^2)^2*(b + a*Cos[e + f*x]^2) ^2) - (-1/2*((a - 7*b)*Cos[e + f*x])/((a + b)*(1 - Cos[e + f*x]^2)*(b + a* Cos[e + f*x]^2)^2) + (((a^2 - 9*a*b + 2*b^2)*Cos[e + f*x])/((a + b)*(b + a *Cos[e + f*x]^2)^2) + (3*(((Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[a] *Cos[e + f*x])/Sqrt[b]])/(Sqrt[a]*(a + b)) - ((a^2 - 10*a*b + 5*b^2)*ArcTa nh[Cos[e + f*x]])/(a + b))/(a + b) + ((a^2 - 6*a*b + b^2)*Cos[e + f*x])/(( a + b)*(b + a*Cos[e + f*x]^2))))/(a + b))/(2*(a + b)))/(4*(a + b)))/f)
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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[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 = 1.72 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a^{2}+2 a b -5 b^{2}\right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (5 a^{2}-10 a b +b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{5}}+\frac {1}{16 \left (a +b \right )^{3} \left (1+\cos \left (f x +e \right )\right )^{2}}-\frac {-3 a +9 b}{16 \left (a +b \right )^{4} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (-3 a^{2}+30 a b -15 b^{2}\right ) \ln \left (1+\cos \left (f x +e \right )\right )}{16 \left (a +b \right )^{5}}-\frac {1}{16 \left (a +b \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{2}}-\frac {-3 a +9 b}{16 \left (a +b \right )^{4} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{16 \left (a +b \right )^{5}}}{f}\) | \(259\) |
| default | \(\frac {\frac {b \left (\frac {\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a^{2}+2 a b -5 b^{2}\right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (5 a^{2}-10 a b +b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{5}}+\frac {1}{16 \left (a +b \right )^{3} \left (1+\cos \left (f x +e \right )\right )^{2}}-\frac {-3 a +9 b}{16 \left (a +b \right )^{4} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (-3 a^{2}+30 a b -15 b^{2}\right ) \ln \left (1+\cos \left (f x +e \right )\right )}{16 \left (a +b \right )^{5}}-\frac {1}{16 \left (a +b \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{2}}-\frac {-3 a +9 b}{16 \left (a +b \right )^{4} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{16 \left (a +b \right )^{5}}}{f}\) | \(259\) |
| risch | \(\text {Expression too large to display}\) | \(1143\) |
Input:
int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/f*(b/(a+b)^5*(((-9/8*a^3-3/4*a^2*b+3/8*a*b^2)*cos(f*x+e)^3-1/8*b*(7*a^2+ 2*a*b-5*b^2)*cos(f*x+e))/(b+a*cos(f*x+e)^2)^2+3/8*(5*a^2-10*a*b+b^2)/(a*b) ^(1/2)*arctan(a*cos(f*x+e)/(a*b)^(1/2)))+1/16/(a+b)^3/(1+cos(f*x+e))^2-1/1 6*(-3*a+9*b)/(a+b)^4/(1+cos(f*x+e))+1/16/(a+b)^5*(-3*a^2+30*a*b-15*b^2)*ln (1+cos(f*x+e))-1/16/(a+b)^3/(-1+cos(f*x+e))^2-1/16*(-3*a+9*b)/(a+b)^4/(-1+ cos(f*x+e))+1/16/(a+b)^5*(3*a^2-30*a*b+15*b^2)*ln(-1+cos(f*x+e)))
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (237) = 474\).
Time = 0.27 (sec) , antiderivative size = 1833, normalized size of antiderivative = 7.13 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")
Output:
[1/16*(6*(a^4 - 5*a^3*b - 5*a^2*b^2 + a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 2 6*a^3*b + 26*a*b^3 - 5*b^4)*cos(f*x + e)^5 - 2*(19*a^3*b - 15*a^2*b^2 - 15 *a*b^3 + 19*b^4)*cos(f*x + e)^3 + 3*((5*a^4 - 10*a^3*b + a^2*b^2)*cos(f*x + e)^8 - 2*(5*a^4 - 15*a^3*b + 11*a^2*b^2 - a*b^3)*cos(f*x + e)^6 + (5*a^4 - 30*a^3*b + 46*a^2*b^2 - 14*a*b^3 + b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 10 *a*b^3 + b^4 + 2*(5*a^3*b - 15*a^2*b^2 + 11*a*b^3 - b^4)*cos(f*x + e)^2)*s qrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos (f*x + e)^2 + b)) - 24*(a^2*b^2 - b^4)*cos(f*x + e) - 3*((a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos (f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e )^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^ 4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 10*a^3*b + 5*a^ 2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e)^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^4)*co s(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5 *a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)...
Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)**5/(a+b*sec(f*x+e)**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (237) = 474\).
Time = 0.12 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.05 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {6 \, {\left (5 \, a^{2} b - 10 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \sqrt {a b}} - \frac {2 \, {\left (3 \, {\left (a^{3} - 6 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{3} - 31 \, a^{2} b + 31 \, a b^{2} - 5 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{2} b - 34 \, a b^{2} + 19 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )}}{{\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (f x + e\right )^{8} - 2 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (f x + e\right )^{6} + a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6} + {\left (a^{6} - 9 \, a^{4} b^{2} - 16 \, a^{3} b^{3} - 9 \, a^{2} b^{4} + b^{6}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b + 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} - 3 \, a b^{5} - b^{6}\right )} \cos \left (f x + e\right )^{2}}}{16 \, f} \] Input:
integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")
Output:
-1/16*(3*(a^2 - 10*a*b + 5*b^2)*log(cos(f*x + e) + 1)/(a^5 + 5*a^4*b + 10* a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) - 3*(a^2 - 10*a*b + 5*b^2)*log(cos(f *x + e) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) - 6 *(5*a^2*b - 10*a*b^2 + b^3)*arctan(a*cos(f*x + e)/sqrt(a*b))/((a^5 + 5*a^4 *b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*sqrt(a*b)) - 2*(3*(a^3 - 6*a ^2*b + a*b^2)*cos(f*x + e)^7 - (5*a^3 - 31*a^2*b + 31*a*b^2 - 5*b^3)*cos(f *x + e)^5 - (19*a^2*b - 34*a*b^2 + 19*b^3)*cos(f*x + e)^3 - 12*(a*b^2 - b^ 3)*cos(f*x + e))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f* x + e)^8 - 2*(a^6 + 3*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5)*c os(f*x + e)^6 + a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6 + (a^6 - 9 *a^4*b^2 - 16*a^3*b^3 - 9*a^2*b^4 + b^6)*cos(f*x + e)^4 + 2*(a^5*b + 3*a^4 *b^2 + 2*a^3*b^3 - 2*a^2*b^4 - 3*a*b^5 - b^6)*cos(f*x + e)^2))/f
Time = 0.24 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.80 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left ({\left | -\cos \left (f x + e\right ) + 1 \right |}\right )}{16 \, {\left (a^{5} f + 5 \, a^{4} b f + 10 \, a^{3} b^{2} f + 10 \, a^{2} b^{3} f + 5 \, a b^{4} f + b^{5} f\right )}} - \frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left ({\left | -\cos \left (f x + e\right ) - 1 \right |}\right )}{16 \, {\left (a^{5} f + 5 \, a^{4} b f + 10 \, a^{3} b^{2} f + 10 \, a^{2} b^{3} f + 5 \, a b^{4} f + b^{5} f\right )}} + \frac {3 \, {\left (5 \, a^{2} b - 10 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{8 \, {\left (a^{5} f + 5 \, a^{4} b f + 10 \, a^{3} b^{2} f + 10 \, a^{2} b^{3} f + 5 \, a b^{4} f + b^{5} f\right )} \sqrt {a b}} + \frac {3 \, a^{3} \cos \left (f x + e\right )^{7} - 18 \, a^{2} b \cos \left (f x + e\right )^{7} + 3 \, a b^{2} \cos \left (f x + e\right )^{7} - 5 \, a^{3} \cos \left (f x + e\right )^{5} + 31 \, a^{2} b \cos \left (f x + e\right )^{5} - 31 \, a b^{2} \cos \left (f x + e\right )^{5} + 5 \, b^{3} \cos \left (f x + e\right )^{5} - 19 \, a^{2} b \cos \left (f x + e\right )^{3} + 34 \, a b^{2} \cos \left (f x + e\right )^{3} - 19 \, b^{3} \cos \left (f x + e\right )^{3} - 12 \, a b^{2} \cos \left (f x + e\right ) + 12 \, b^{3} \cos \left (f x + e\right )}{8 \, {\left (a^{4} f + 4 \, a^{3} b f + 6 \, a^{2} b^{2} f + 4 \, a b^{3} f + b^{4} f\right )} {\left (a \cos \left (f x + e\right )^{4} - a \cos \left (f x + e\right )^{2} + b \cos \left (f x + e\right )^{2} - b\right )}^{2}} \] Input:
integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")
Output:
3/16*(a^2 - 10*a*b + 5*b^2)*log(abs(-cos(f*x + e) + 1))/(a^5*f + 5*a^4*b*f + 10*a^3*b^2*f + 10*a^2*b^3*f + 5*a*b^4*f + b^5*f) - 3/16*(a^2 - 10*a*b + 5*b^2)*log(abs(-cos(f*x + e) - 1))/(a^5*f + 5*a^4*b*f + 10*a^3*b^2*f + 10 *a^2*b^3*f + 5*a*b^4*f + b^5*f) + 3/8*(5*a^2*b - 10*a*b^2 + b^3)*arctan(a* cos(f*x + e)/sqrt(a*b))/((a^5*f + 5*a^4*b*f + 10*a^3*b^2*f + 10*a^2*b^3*f + 5*a*b^4*f + b^5*f)*sqrt(a*b)) + 1/8*(3*a^3*cos(f*x + e)^7 - 18*a^2*b*cos (f*x + e)^7 + 3*a*b^2*cos(f*x + e)^7 - 5*a^3*cos(f*x + e)^5 + 31*a^2*b*cos (f*x + e)^5 - 31*a*b^2*cos(f*x + e)^5 + 5*b^3*cos(f*x + e)^5 - 19*a^2*b*co s(f*x + e)^3 + 34*a*b^2*cos(f*x + e)^3 - 19*b^3*cos(f*x + e)^3 - 12*a*b^2* cos(f*x + e) + 12*b^3*cos(f*x + e))/((a^4*f + 4*a^3*b*f + 6*a^2*b^2*f + 4* a*b^3*f + b^4*f)*(a*cos(f*x + e)^4 - a*cos(f*x + e)^2 + b*cos(f*x + e)^2 - b)^2)
Time = 17.13 (sec) , antiderivative size = 5613, normalized size of antiderivative = 21.84 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(sin(e + f*x)^5*(a + b/cos(e + f*x)^2)^3),x)
Output:
(atan(((((cos(e + f*x)*(9*a*b^6 - 180*a^6*b + 9*a^7 - 180*a^2*b^5 + 1215*a ^3*b^4 - 1800*a^4*b^3 + 1215*a^5*b^2))/(32*(8*a*b^7 + 8*a^7*b + a^8 + b^8 + 28*a^2*b^6 + 56*a^3*b^5 + 70*a^4*b^4 + 56*a^5*b^3 + 28*a^6*b^2)) + (3*(- a*b)^(1/2)*((6*a^13*b - 6*a^2*b^12 - 54*a^3*b^11 - 210*a^4*b^10 - 450*a^5* b^9 - 540*a^6*b^8 - 252*a^7*b^7 + 252*a^8*b^6 + 540*a^9*b^5 + 450*a^10*b^4 + 210*a^11*b^3 + 54*a^12*b^2)/(12*a*b^11 + 12*a^11*b + a^12 + b^12 + 66*a ^2*b^10 + 220*a^3*b^9 + 495*a^4*b^8 + 792*a^5*b^7 + 924*a^6*b^6 + 792*a^7* b^5 + 495*a^8*b^4 + 220*a^9*b^3 + 66*a^10*b^2) - (3*cos(e + f*x)*(-a*b)^(1 /2)*(5*a^2 - 10*a*b + b^2)*(2304*a^12*b + 256*a^13 - 256*a^2*b^11 - 2304*a ^3*b^10 - 8960*a^4*b^9 - 19200*a^5*b^8 - 23040*a^6*b^7 - 10752*a^7*b^6 + 1 0752*a^8*b^5 + 23040*a^9*b^4 + 19200*a^10*b^3 + 8960*a^11*b^2))/(512*(a*b^ 5 + 5*a^5*b + a^6 + 5*a^2*b^4 + 10*a^3*b^3 + 10*a^4*b^2)*(8*a*b^7 + 8*a^7* b + a^8 + b^8 + 28*a^2*b^6 + 56*a^3*b^5 + 70*a^4*b^4 + 56*a^5*b^3 + 28*a^6 *b^2)))*(5*a^2 - 10*a*b + b^2))/(16*(a*b^5 + 5*a^5*b + a^6 + 5*a^2*b^4 + 1 0*a^3*b^3 + 10*a^4*b^2)))*(-a*b)^(1/2)*(5*a^2 - 10*a*b + b^2)*3i)/(16*(a*b ^5 + 5*a^5*b + a^6 + 5*a^2*b^4 + 10*a^3*b^3 + 10*a^4*b^2)) + (((cos(e + f* x)*(9*a*b^6 - 180*a^6*b + 9*a^7 - 180*a^2*b^5 + 1215*a^3*b^4 - 1800*a^4*b^ 3 + 1215*a^5*b^2))/(32*(8*a*b^7 + 8*a^7*b + a^8 + b^8 + 28*a^2*b^6 + 56*a^ 3*b^5 + 70*a^4*b^4 + 56*a^5*b^3 + 28*a^6*b^2)) - (3*(-a*b)^(1/2)*((6*a^13* b - 6*a^2*b^12 - 54*a^3*b^11 - 210*a^4*b^10 - 450*a^5*b^9 - 540*a^6*b^8...
Time = 0.36 (sec) , antiderivative size = 1988, normalized size of antiderivative = 7.74 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x)
Output:
( - 60*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt( b))*sin(e + f*x)**8*a**4 + 120*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**8*a**3*b - 12*sqrt(b)*sqrt(a)*at an((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**8*a**2* b**2 + 120*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/s qrt(b))*sin(e + f*x)**6*a**4 - 120*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan(( e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**6*a**3*b - 216*sqrt(b)*sqrt( a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**6* a**2*b**2 + 24*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a ))/sqrt(b))*sin(e + f*x)**6*a*b**3 - 60*sqrt(b)*sqrt(a)*atan((sqrt(a + b)* tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**4*a**4 + 168*sqrt(b)*sq rt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)* *4*a**2*b**2 + 96*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqr t(a))/sqrt(b))*sin(e + f*x)**4*a*b**3 - 12*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**4*b**4 + 60*sqrt(b)* sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f*x )**8*a**4 - 120*sqrt(b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt( a))/sqrt(b))*sin(e + f*x)**8*a**3*b + 12*sqrt(b)*sqrt(a)*atan((sqrt(a + b) *tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f*x)**8*a**2*b**2 - 120*sqrt (b)*sqrt(a)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(...