\(\int \csc ^3(e+f x) (a+b \sec ^2(e+f x))^2 \, dx\) [19]

Optimal result

Integrand size = 23, antiderivative size = 104 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {(a+b) (a+5 b) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {\left (3 a^2+6 a b+5 b^2\right ) \cot (e+f x) \csc (e+f x)}{6 f}+\frac {b (6 a+5 b) \sec (e+f x)}{3 f}+\frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f} \]

[Out]

-1/2*(a+b)*(a+5*b)*arctanh(cos(f*x+e))/f-1/6*(3*a^2+6*a*b+5*b^2)*cot(f*x+e)*csc(f*x+e)/f+1/3*b*(6*a+5*b)*sec(f
*x+e)/f+1/3*b^2*csc(f*x+e)^2*sec(f*x+e)^3/f

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4218, 473, 467, 464, 212} \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (3 a^2+6 a b+5 b^2\right ) \cot (e+f x) \csc (e+f x)}{6 f}-\frac {(a+b) (a+5 b) \text {arctanh}(\cos (e+f x))}{2 f}+\frac {b (6 a+5 b) \sec (e+f x)}{3 f}+\frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f} \]

[In]

Int[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]

[Out]

-1/2*((a + b)*(a + 5*b)*ArcTanh[Cos[e + f*x]])/f - ((3*a^2 + 6*a*b + 5*b^2)*Cot[e + f*x]*Csc[e + f*x])/(6*f) +
(b*(6*a + 5*b)*Sec[e + f*x])/(3*f) + (b^2*Csc[e + f*x]^2*Sec[e + f*x]^3)/(3*f)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 467

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] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[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])

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[c^2*(e*x)^(m
+ 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 4218

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, Dist[-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
]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^2}{x^4 \left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f}-\frac {\text {Subst}\left (\int \frac {b (6 a+5 b)+3 a^2 x^2}{x^2 \left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{3 f} \\ & = -\frac {\left (3 a^2+6 a b+5 b^2\right ) \cot (e+f x) \csc (e+f x)}{6 f}+\frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f}+\frac {\text {Subst}\left (\int \frac {-2 b (6 a+5 b)-\left (3 a^2+6 a b+5 b^2\right ) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\cos (e+f x)\right )}{6 f} \\ & = -\frac {\left (3 a^2+6 a b+5 b^2\right ) \cot (e+f x) \csc (e+f x)}{6 f}+\frac {b (6 a+5 b) \sec (e+f x)}{3 f}+\frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f}-\frac {((a+b) (a+5 b)) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (e+f x)\right )}{2 f} \\ & = -\frac {(a+b) (a+5 b) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {\left (3 a^2+6 a b+5 b^2\right ) \cot (e+f x) \csc (e+f x)}{6 f}+\frac {b (6 a+5 b) \sec (e+f x)}{3 f}+\frac {b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1021\) vs. \(2(104)=208\).

Time = 8.09 (sec) , antiderivative size = 1021, normalized size of antiderivative = 9.82 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (-a^2-2 a b-b^2\right ) \cos ^4(e+f x) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (a+b \sec ^2(e+f x)\right )^2}{2 f (a+2 b+a \cos (2 e+2 f x))^2}-\frac {2 \left (a^2+6 a b+5 b^2\right ) \cos ^4(e+f x) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) \left (a+b \sec ^2(e+f x)\right )^2}{f (a+2 b+a \cos (2 e+2 f x))^2}+\frac {2 \left (a^2+6 a b+5 b^2\right ) \cos ^4(e+f x) \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) \left (a+b \sec ^2(e+f x)\right )^2}{f (a+2 b+a \cos (2 e+2 f x))^2}+\frac {2 b (12 a+13 b) \cos ^4(e+f x) \sec (e) \left (a+b \sec ^2(e+f x)\right )^2}{3 f (a+2 b+a \cos (2 e+2 f x))^2}+\frac {\left (a^2+2 a b+b^2\right ) \cos ^4(e+f x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (a+b \sec ^2(e+f x)\right )^2}{2 f (a+2 b+a \cos (2 e+2 f x))^2}+\frac {2 b^2 \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \sin \left (\frac {f x}{2}\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^3}+\frac {\cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \left (b^2 \cos \left (\frac {e}{2}\right )+b^2 \sin \left (\frac {e}{2}\right )\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {2 \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \left (12 a b \sin \left (\frac {f x}{2}\right )+13 b^2 \sin \left (\frac {f x}{2}\right )\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}-\frac {2 b^2 \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \sin \left (\frac {f x}{2}\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^3}+\frac {\cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \left (b^2 \cos \left (\frac {e}{2}\right )-b^2 \sin \left (\frac {e}{2}\right )\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {2 \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \left (12 a b \sin \left (\frac {f x}{2}\right )+13 b^2 \sin \left (\frac {f x}{2}\right )\right )}{3 f (a+2 b+a \cos (2 e+2 f x))^2 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )} \]

[In]

Integrate[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]

[Out]

((-a^2 - 2*a*b - b^2)*Cos[e + f*x]^4*Csc[e/2 + (f*x)/2]^2*(a + b*Sec[e + f*x]^2)^2)/(2*f*(a + 2*b + a*Cos[2*e
+ 2*f*x])^2) - (2*(a^2 + 6*a*b + 5*b^2)*Cos[e + f*x]^4*Log[Cos[e/2 + (f*x)/2]]*(a + b*Sec[e + f*x]^2)^2)/(f*(a
+ 2*b + a*Cos[2*e + 2*f*x])^2) + (2*(a^2 + 6*a*b + 5*b^2)*Cos[e + f*x]^4*Log[Sin[e/2 + (f*x)/2]]*(a + b*Sec[e
+ f*x]^2)^2)/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (2*b*(12*a + 13*b)*Cos[e + f*x]^4*Sec[e]*(a + b*Sec[e + f
*x]^2)^2)/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + ((a^2 + 2*a*b + b^2)*Cos[e + f*x]^4*Sec[e/2 + (f*x)/2]^2*(a
+ b*Sec[e + f*x]^2)^2)/(2*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (2*b^2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^
2*Sin[(f*x)/2])/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f
*x)/2])^3) + (Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(b^2*Cos[e/2] + b^2*Sin[e/2]))/(3*f*(a + 2*b + a*Cos[2*e
+ 2*f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^2) + (2*Cos[e + f*x]^4*(a + b*Sec
[e + f*x]^2)^2*(12*a*b*Sin[(f*x)/2] + 13*b^2*Sin[(f*x)/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] -
Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) - (2*b^2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*Sin[(f*x
)/2])/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^3)
+ (Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(b^2*Cos[e/2] - b^2*Sin[e/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])
^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) - (2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^
2)^2*(12*a*b*Sin[(f*x)/2] + 13*b^2*Sin[(f*x)/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] + Sin[e/2])*
(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\frac {1}{2 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3}{2 \cos \left (f x +e \right )}+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+b^{2} \left (\frac {1}{3 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{3}}-\frac {5}{6 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {5}{2 \cos \left (f x +e \right )}+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )}{f}\)	\(163\)
default	\(\frac {a^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\frac {1}{2 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3}{2 \cos \left (f x +e \right )}+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+b^{2} \left (\frac {1}{3 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{3}}-\frac {5}{6 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {5}{2 \cos \left (f x +e \right )}+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )}{f}\)	\(163\)
norman	\(\frac {\frac {a^{2}+2 a b +b^{2}}{8 f}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{8 f}-\frac {\left (7 a^{2}+46 a b +55 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{8 f}-\frac {\left (9 a^{2}+66 a b +65 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{12 f}+\frac {\left (11 a^{2}+86 a b +75 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {\left (a^{2}+6 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\)	\(194\)
parallelrisch	\(\frac {\frac {3 \left (\frac {\cos \left (3 f x +3 e \right )}{3}+\cos \left (f x +e \right )\right ) \left (a +5 b \right ) \left (a +b \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {3 \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\left (a^{2}+\frac {22}{3} a b +\frac {65}{9} b^{2}\right ) \cos \left (3 f x +3 e \right )+\frac {32 \left (a^{2}+2 a b +\frac {5}{3} b^{2}\right ) \cos \left (2 f x +2 e \right )}{3}+\frac {8 \left (a +5 b \right ) \left (a +b \right ) \cos \left (4 f x +4 e \right )}{3}+\left (a^{2}+\frac {22}{3} a b +\frac {65}{9} b^{2}\right ) \cos \left (5 f x +5 e \right )+2 \left (-a^{2}-\frac {22}{3} a b -\frac {65}{9} b^{2}\right ) \cos \left (f x +e \right )+8 a^{2}+\frac {16 a b}{3}-\frac {88 b^{2}}{9}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{128}}{f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\)	\(216\)
risch	\(\frac {{\mathrm e}^{i \left (f x +e \right )} \left (3 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+18 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+15 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+12 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+24 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+20 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+18 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}+12 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-22 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+12 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+24 a b \,{\mathrm e}^{2 i \left (f x +e \right )}+20 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 a^{2}+18 a b +15 b^{2}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) a^{2}}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) a b}{f}-\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{2 f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) a^{2}}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) a b}{f}+\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{2 f}\)	\(367\)

[In]

int(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(a^2*(-1/2*csc(f*x+e)*cot(f*x+e)+1/2*ln(csc(f*x+e)-cot(f*x+e)))+2*a*b*(-1/2/sin(f*x+e)^2/cos(f*x+e)+3/2/co
s(f*x+e)+3/2*ln(csc(f*x+e)-cot(f*x+e)))+b^2*(1/3/sin(f*x+e)^2/cos(f*x+e)^3-5/6/sin(f*x+e)^2/cos(f*x+e)+5/2/cos
(f*x+e)+5/2*ln(csc(f*x+e)-cot(f*x+e))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (96) = 192\).

Time = 0.26 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.86 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {6 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 4 \, {\left (6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, b^{2} - 3 \, {\left ({\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{12 \, {\left (f \cos \left (f x + e\right )^{5} - f \cos \left (f x + e\right )^{3}\right )}} \]

[In]

integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

1/12*(6*(a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^4 - 4*(6*a*b + 5*b^2)*cos(f*x + e)^2 - 4*b^2 - 3*((a^2 + 6*a*b + 5*
b^2)*cos(f*x + e)^5 - (a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^2 + 6*a*b + 5*
b^2)*cos(f*x + e)^5 - (a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^3)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^5 -
f*cos(f*x + e)^3)

Sympy [F]

\[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \csc ^{3}{\left (e + f x \right )}\, dx \]

[In]

integrate(csc(f*x+e)**3*(a+b*sec(f*x+e)**2)**2,x)

[Out]

Integral((a + b*sec(e + f*x)**2)**2*csc(e + f*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - 3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (6 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, b^{2}\right )}}{\cos \left (f x + e\right )^{5} - \cos \left (f x + e\right )^{3}}}{12 \, f} \]

[In]

integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

-1/12*(3*(a^2 + 6*a*b + 5*b^2)*log(cos(f*x + e) + 1) - 3*(a^2 + 6*a*b + 5*b^2)*log(cos(f*x + e) - 1) - 2*(3*(a
^2 + 6*a*b + 5*b^2)*cos(f*x + e)^4 - 2*(6*a*b + 5*b^2)*cos(f*x + e)^2 - 2*b^2)/(cos(f*x + e)^5 - cos(f*x + e)^
3))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (96) = 192\).

Time = 0.35 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.28 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\frac {3 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {6 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {3 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - 6 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right ) - \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2} - \frac {2 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {12 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {10 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}{\cos \left (f x + e\right ) - 1} - \frac {16 \, {\left (6 \, a b + 7 \, b^{2} + \frac {12 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {12 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {6 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1\right )}^{3}}}{24 \, f} \]

[In]

integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")

[Out]

-1/24*(3*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 6*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 3*b^2*(cos(
f*x + e) - 1)/(cos(f*x + e) + 1) - 6*(a^2 + 6*a*b + 5*b^2)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1)) -
3*(a^2 + 2*a*b + b^2 - 2*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 12*a*b*(cos(f*x + e) - 1)/(cos(f*x + e)
+ 1) - 10*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1))*(cos(f*x + e) + 1)/(cos(f*x + e) - 1) - 16*(6*a*b + 7*b^2
+ 12*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 12*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 6*a*b*(cos(f*
x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 9*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/((cos(f*x + e) - 1)/(cos
(f*x + e) + 1) + 1)^3)/f

Mupad [B] (verification not implemented)

Time = 18.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\frac {b^2}{3}+{\cos \left (e+f\,x\right )}^2\,\left (\frac {5\,b^2}{3}+2\,a\,b\right )-{\cos \left (e+f\,x\right )}^4\,\left (\frac {a^2}{2}+3\,a\,b+\frac {5\,b^2}{2}\right )}{f\,\left ({\cos \left (e+f\,x\right )}^3-{\cos \left (e+f\,x\right )}^5\right )}-\frac {\mathrm {atanh}\left (\cos \left (e+f\,x\right )\right )\,\left (a+b\right )\,\left (a+5\,b\right )}{2\,f} \]

[In]

int((a + b/cos(e + f*x)^2)^2/sin(e + f*x)^3,x)

[Out]

(b^2/3 + cos(e + f*x)^2*(2*a*b + (5*b^2)/3) - cos(e + f*x)^4*(3*a*b + a^2/2 + (5*b^2)/2))/(f*(cos(e + f*x)^3 -
cos(e + f*x)^5)) - (atanh(cos(e + f*x))*(a + b)*(a + 5*b))/(2*f)