\(\int \frac {\csc ^5(e+f x)}{(a+b \tan ^2(e+f x))^3} \, dx\) [85]

Optimal result

Integrand size = 23, antiderivative size = 259 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 \sqrt {a-b} f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^5 f}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )} \]

[Out]

-3/8*(a^2-12*a*b+16*b^2)*arctanh(cos(f*x+e))/a^5/f-1/8*(5*a-8*b)*cot(f*x+e)*csc(f*x+e)/a^2/f/(a-b+b*sec(f*x+e)
^2)^2-1/4*cot(f*x+e)^3*csc(f*x+e)/a/f/(a-b+b*sec(f*x+e)^2)^2-1/8*(7*a-12*b)*b*sec(f*x+e)/a^3/f/(a-b+b*sec(f*x+
e)^2)^2-3/2*(a-2*b)*b*sec(f*x+e)/a^4/f/(a-b+b*sec(f*x+e)^2)-3/8*(5*a^2-20*a*b+16*b^2)*arctan(sec(f*x+e)*b^(1/2
)/(a-b)^(1/2))*b^(1/2)/a^5/f/(a-b)^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3745, 481, 541, 536, 213, 211} \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac {b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 f \sqrt {a-b}}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^5 f}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \]

[In]

Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

(-3*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^5*Sqrt[a - b]*f) - (3*(
a^2 - 12*a*b + 16*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^5*f) - ((5*a - 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a
- b + b*Sec[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^2) - ((7*a - 12*b
)*b*Sec[e + f*x])/(8*a^3*f*(a - b + b*Sec[e + f*x]^2)^2) - (3*(a - 2*b)*b*Sec[e + f*x])/(2*a^4*f*(a - b + b*Se
c[e + f*x]^2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 481

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

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3745

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-a+b+(-4 a+7 b) x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{4 a f} \\ & = -\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-((3 a-8 b) (a-b))+5 (5 a-8 b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{8 a^2 f} \\ & = -\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-12 (a-4 b) (a-b)^2+12 (7 a-12 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{32 a^3 (a-b) f} \\ & = -\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 (a-b)^2 \left (a^2-8 a b+8 b^2\right )+96 (a-2 b) (a-b)^2 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{64 a^4 (a-b)^2 f} \\ & = -\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\left (3 b \left (5 a^2-20 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{8 a^5 f}+\frac {\left (3 \left (a^2-12 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 a^5 f} \\ & = -\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 \sqrt {a-b} f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^5 f}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}-\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}+\frac {b^2 \cos (e+f x)}{a^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {3 \left (3 a b \cos (e+f x)-4 b^2 \cos (e+f x)\right )}{4 a^4 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}-\frac {3 (a-4 b) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}-\frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 (a-4 b) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}+\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f} \]

[In]

Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

(-3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqr
t[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a^5*(-a + b)*f) - (3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*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^5*(-a + b)*f) + (
b^2*Cos[e + f*x])/(a^3*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (3*(3*a*b*Cos[e + f*x] - 4*b^2
*Cos[e + f*x]))/(4*a^4*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) - (3*(a - 4*b)*Csc[(e + f*x)/2]^2)
/(32*a^4*f) - Csc[(e + f*x)/2]^4/(64*a^3*f) - (3*(a^2 - 12*a*b + 16*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^5*f) + (3
*(a^2 - 12*a*b + 16*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^5*f) + (3*(a - 4*b)*Sec[(e + f*x)/2]^2)/(32*a^4*f) + Sec[
(e + f*x)/2]^4/(64*a^3*f)

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}+\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}}{f}\)	\(265\)
default	\(\frac {\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}+\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}}{f}\)	\(265\)
risch	\(\text {Expression too large to display}\)	\(1036\)

[In]

int(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(b/a^5*((-3/8*a*(3*a^2-7*a*b+4*b^2)*cos(f*x+e)^3+(-7/8*a^2*b+3/2*a*b^2)*cos(f*x+e))/(a*cos(f*x+e)^2-b*cos(
f*x+e)^2+b)^2+3/8*(5*a^2-20*a*b+16*b^2)/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2)))+1/16/a^3/(co
s(f*x+e)+1)^2-1/16*(-3*a+12*b)/a^4/(cos(f*x+e)+1)+1/16/a^5*(-3*a^2+36*a*b-48*b^2)*ln(cos(f*x+e)+1)-1/16/a^3/(c
os(f*x+e)-1)^2-1/16*(-3*a+12*b)/a^4/(cos(f*x+e)-1)+1/16/a^5*(3*a^2-36*a*b+48*b^2)*ln(cos(f*x+e)-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (239) = 478\).

Time = 0.46 (sec) , antiderivative size = 1693, normalized size of antiderivative = 6.54 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(6*(a^4 - 9*a^3*b + 16*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 46*a^3*b + 108*a^2*b^2 - 72*a*b^3)
*cos(f*x + e)^5 - 2*(19*a^3*b - 72*a^2*b^2 + 72*a*b^3)*cos(f*x + e)^3 + 3*((5*a^4 - 30*a^3*b + 61*a^2*b^2 - 52
*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(5*a^4 - 35*a^3*b + 86*a^2*b^2 - 88*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (5*a^
4 - 50*a^3*b + 166*a^2*b^2 - 216*a*b^3 + 96*b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 20*a*b^3 + 16*b^4 + 2*(5*a^3*b -
30*a^2*b^2 + 56*a*b^3 - 32*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)) - 24*(a^2*b^2 - 2*a*b^3)*cos(f*x + e) - 3*((a^4 -
14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^
4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 1
6*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 14*
a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*c
os(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^
4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 2*a^6*b +
a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a
^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2), 1/16*(6*(a^4 - 9*a^3*b + 16*a^2*b^2 - 8*a*
b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 46*a^3*b + 108*a^2*b^2 - 72*a*b^3)*cos(f*x + e)^5 - 2*(19*a^3*b - 72*a^2*b^2
+ 72*a*b^3)*cos(f*x + e)^3 - 6*((5*a^4 - 30*a^3*b + 61*a^2*b^2 - 52*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(5*a^4
- 35*a^3*b + 86*a^2*b^2 - 88*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (5*a^4 - 50*a^3*b + 166*a^2*b^2 - 216*a*b^3 + 96
*b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 20*a*b^3 + 16*b^4 + 2*(5*a^3*b - 30*a^2*b^2 + 56*a*b^3 - 32*b^4)*cos(f*x +
e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - 24*(a^2*b^2 - 2*a*b^3)*cos(f*x + e) -
3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^
3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*
a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((
a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 +
32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^
3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 -
2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^
6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(csc(f*x+e)**5/(a+b*tan(f*x+e)**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (239) = 478\).

Time = 0.91 (sec) , antiderivative size = 861, normalized size of antiderivative = 3.32 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/64*(12*(a^2 - 12*a*b + 16*b^2)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/a^5 - 24*(5*a^2*b - 20*a*b^
2 + 16*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)))/(s
qrt(a*b - b^2)*a^5) - (8*a^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 24*a^2*b*(cos(f*x + e) - 1)/(cos(f*x + e)
+ 1) - a^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/a^6 - (a^4 - 4*a^4*(cos(f*x + e) - 1)/(cos(f*x + e) + 1
) + 16*a^3*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 20*a^4*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 216*a^
3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 304*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 20*a^4
*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 360*a^3*b*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 1024*a^2*b^
2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 896*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 5*a^4*(cos
(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 + 64*a^3*b*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 - 192*a^2*b^2*(cos(
f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 + 256*a*b^3*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 - 256*b^4*(cos(f*x
+ e) - 1)^4/(cos(f*x + e) + 1)^4 + 16*a^4*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 168*a^3*b*(cos(f*x + e)
- 1)^5/(cos(f*x + e) + 1)^5 + 384*a^2*b^2*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 256*a*b^3*(cos(f*x + e)
- 1)^5/(cos(f*x + e) + 1)^5 + 6*a^4*(cos(f*x + e) - 1)^6/(cos(f*x + e) + 1)^6 - 72*a^3*b*(cos(f*x + e) - 1)^6/
(cos(f*x + e) + 1)^6 + 96*a^2*b^2*(cos(f*x + e) - 1)^6/(cos(f*x + e) + 1)^6)/(a^5*(a*(cos(f*x + e) - 1)/(cos(f
*x + e) + 1) + 2*a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 4*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 +
a*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)^2))/f

Mupad [B] (verification not implemented)

Time = 11.94 (sec) , antiderivative size = 1357, normalized size of antiderivative = 5.24 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(1/(sin(e + f*x)^5*(a + b*tan(e + f*x)^2)^3),x)

[Out]

tan(e/2 + (f*x)/2)^4/(64*a^3*f) + (tan(e/2 + (f*x)/2)^2*((3*(a - 2*b))/(16*a^4) - 1/(16*a^3)))/f + (tan(e/2 +
(f*x)/2)^4*(100*a*b^2 - 72*a^2*b + (13*a^3)/2) - tan(e/2 + (f*x)/2)^10*(144*a*b^2 - 42*a^2*b + 2*a^3 - 128*b^3
) - tan(e/2 + (f*x)/2)^6*(496*a*b^2 - 174*a^2*b + 11*a^3 - 416*b^3) + tan(e/2 + (f*x)/2)^2*(4*a^2*b - a^3) - a
^3/4 + (tan(e/2 + (f*x)/2)^8*(31*a^4 - 592*a^3*b - 2944*a*b^3 + 1792*b^4 + 2016*a^2*b^2))/(4*a))/(f*(16*a^6*ta
n(e/2 + (f*x)/2)^4 + 16*a^6*tan(e/2 + (f*x)/2)^12 + tan(e/2 + (f*x)/2)^8*(96*a^6 - 256*a^5*b + 256*a^4*b^2) +
tan(e/2 + (f*x)/2)^6*(128*a^5*b - 64*a^6) + tan(e/2 + (f*x)/2)^10*(128*a^5*b - 64*a^6))) + (log(tan(e/2 + (f*x
)/2))*(3*a^2 - 36*a*b + 48*b^2))/(8*a^5*f) + (3*b^(1/2)*atan((a^13*(a - b)^(3/2)*((256*((3456*b^8 - 11232*a*b^
7 + 14256*a^2*b^6 - 8910*a^3*b^5 + 2835*a^4*b^4 - (3375*a^5*b^3)/8 + (675*a^6*b^2)/32)/a^12 - (9*b*(5*a^2 - 20
*a*b + 16*b^2)^2*(192*a^14 - 4992*a^13*b + 24576*a^10*b^4 - 43008*a^11*b^3 + 24576*a^12*b^2))/(8192*a^22*(a -
b)))*(1728*a*b^4 - 45*a^4*b + a^5 - 768*b^5 - 1344*a^2*b^3 + 420*a^3*b^2))/(b^(1/2)*(b*(b*(b*(1680*a^7 + b*(76
8*a^5*b - 1920*a^6)) - 600*a^8) + 75*a^9) - 4*a^10)) - 256*tan(e/2 + (f*x)/2)^2*((((4752*a*b^6 - 1728*b^7 - 48
60*a^2*b^5 + 2295*a^3*b^4 - (2025*a^4*b^3)/4 + (675*a^5*b^2)/16)/a^11 + (9*b*(5*a^2 - 20*a*b + 16*b^2)^2*(3552
*a^12*b - 96*a^13 + 73728*a^8*b^5 - 165888*a^9*b^4 + 125952*a^10*b^3 - 36480*a^11*b^2))/(4096*a^21*(a - b)))*(
1728*a*b^4 - 45*a^4*b + a^5 - 768*b^5 - 1344*a^2*b^3 + 420*a^3*b^2))/(b^(1/2)*(b*(b*(b*(1680*a^7 + b*(768*a^5*
b - 1920*a^6)) - 600*a^8) + 75*a^9) - 4*a^10)) - (((b^(3/2)*(5*a^2 - 20*a*b + 16*b^2)^3*((351*a^15*b)/128 - (2
7*a^16)/256 + (81*a^13*b^3)/16 - (243*a^14*b^2)/32))/(a^26*(a - b)^(3/2)) - (3*b^(1/2)*(5*a^2 - 20*a*b + 16*b^
2)*(540*a^9*b + 110592*a^3*b^7 - 331776*a^4*b^6 + 389376*a^5*b^5 - 225792*a^6*b^4 + 67248*a^7*b^3 - 9720*a^8*b
^2))/(256*a^16*(a - b)^(1/2)))*(4224*a*b^4 - 330*a^4*b + 17*a^5 - 1536*b^5 - 4224*a^2*b^3 + 1848*a^3*b^2))/(2*
a^5*(a - b)^(1/2)*(1920*a*b^4 - 75*a^4*b + 4*a^5 - 768*b^5 - 1680*a^2*b^3 + 600*a^3*b^2))) + (128*((b^(3/2)*(5
*a^2 - 20*a*b + 16*b^2)^3*((27*a^17)/256 - (27*a^16*b)/64 + (27*a^15*b^2)/64))/(a^27*(a - b)^(3/2)) + (3*b^(1/
2)*(5*a^2 - 20*a*b + 16*b^2)*(720*a^10*b - 92160*a^5*b^6 + 230400*a^6*b^5 - 210816*a^7*b^4 + 85824*a^8*b^3 - 1
4760*a^9*b^2))/(512*a^17*(a - b)^(1/2)))*(4224*a*b^4 - 330*a^4*b + 17*a^5 - 1536*b^5 - 4224*a^2*b^3 + 1848*a^3
*b^2))/(a^5*(a - b)^(1/2)*(1920*a*b^4 - 75*a^4*b + 4*a^5 - 768*b^5 - 1680*a^2*b^3 + 600*a^3*b^2))))/(675*a^4*b
- 17280*a*b^4 + 6912*b^5 + 15120*a^2*b^3 - 5400*a^3*b^2))*(5*a^2 - 20*a*b + 16*b^2))/(8*a^5*f*(a - b)^(1/2))