3.145 \(\int \frac {\csc ^5(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=237 \[ -\frac {5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {\left (3 a^2-30 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 a^{9/2} f}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]

[Out]

-1/8*(3*a^2-30*a*b+35*b^2)*arctanh(sec(f*x+e)*a^(1/2)/(a-b+b*sec(f*x+e)^2)^(1/2))/a^(9/2)/f-1/8*(5*a-7*b)*cot(
f*x+e)*csc(f*x+e)/a^2/f/(a-b+b*sec(f*x+e)^2)^(3/2)-1/4*cot(f*x+e)^3*csc(f*x+e)/a/f/(a-b+b*sec(f*x+e)^2)^(3/2)-
1/24*(23*a-35*b)*b*sec(f*x+e)/a^3/f/(a-b+b*sec(f*x+e)^2)^(3/2)-5/24*(11*a-21*b)*b*sec(f*x+e)/a^4/f/(a-b+b*sec(
f*x+e)^2)^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3664, 470, 527, 12, 377, 207} \[ -\frac {\left (3 a^2-30 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 a^{9/2} f}-\frac {5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a^2 - 30*a*b + 35*b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(9/2)*f) - ((
5*a - 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (Cot[e + f*x]^3*Csc[e + f*x
])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((23*a - 35*b)*b*Sec[e + f*x])/(24*a^3*f*(a - b + b*Sec[e + f*x]
^2)^(3/2)) - (5*(11*a - 21*b)*b*Sec[e + f*x])/(24*a^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 207

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 470

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 527

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 3664

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*} \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a-b+b x^2\right )^{5/2}} \, 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 )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-a+b-2 (2 a-3 b) x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{4 a f}\\ &=-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-(3 a-7 b) (a-b)+4 (5 a-7 b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{8 a^2 f}\\ &=-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-(9 a-35 b) (a-b)^2+2 (23 a-35 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{24 a^3 (a-b) f}\\ &=-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {\operatorname {Subst}\left (\int -\frac {3 (a-b)^2 \left (3 a^2-30 a b+35 b^2\right )}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{24 a^4 (a-b)^2 f}\\ &=-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}+\frac {\left (3 a^2-30 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a^4 f}\\ &=-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}+\frac {\left (3 a^2-30 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+a x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^4 f}\\ &=-\frac {\left (3 a^2-30 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^{9/2} f}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}\\ \end {align*}

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Mathematica [B]  time = 6.98, size = 1142, normalized size = 4.82 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((4*b^2*Cos[e + f*x])/(3*a^3*(
a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (2*(2*a*b*Cos[e + f*x] - 3*b^2*Cos[e + f*x]))/(a^4*(a +
b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) + ((-3*a*Cos[e + f*x] + 11*b*Cos[e + f*x])*Csc[e + f*x]^2)/(8*a^
4) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*a^3)))/f + ((3*a^2 - 30*a*b + 35*b^2)*(-1/4*((2*Sqrt[a]*ArcTanh[(Sqrt[b]
*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[b]*(ArcTanh[(a
- a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)
/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(
e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a -
b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[
(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a
- b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2
]^2)^2]) + ((2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e
+ f*x)/2]^2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan
[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt
[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1
+ Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*
(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2
)^2])/(4*Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan
[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])))/(8*a^4*f)

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fricas [B]  time = 0.80, size = 1037, normalized size = 4.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*((3*a^4 - 36*a^3*b + 98*a^2*b^2 - 100*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(3*a^4 - 39*a^3*b + 131*a^2*
b^2 - 165*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (3*a^4 - 48*a^3*b + 233*a^2*b^2 - 390*a*b^3 + 210*b^4)*cos(f*x + e)
^4 + 3*a^2*b^2 - 30*a*b^3 + 35*b^4 + 2*(3*a^3*b - 36*a^2*b^2 + 95*a*b^3 - 70*b^4)*cos(f*x + e)^2)*sqrt(a)*log(
-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)
/(cos(f*x + e)^2 - 1)) + 2*(3*(3*a^4 - 33*a^3*b + 65*a^2*b^2 - 35*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 177*a^3*b
+ 445*a^2*b^2 - 315*a*b^3)*cos(f*x + e)^5 - (78*a^3*b - 305*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^3 - 5*(11*a^2*b^
2 - 21*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 2*a^6*b + a^5*b^2)*f*co
s(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), 1/24*(3*((3*a^4 - 36*a^3*b + 98*a^2*b^2 - 100*a*b^3 + 35
*b^4)*cos(f*x + e)^8 - 2*(3*a^4 - 39*a^3*b + 131*a^2*b^2 - 165*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (3*a^4 - 48*a^
3*b + 233*a^2*b^2 - 390*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 30*a*b^3 + 35*b^4 + 2*(3*a^3*b - 36*a^2*
b^2 + 95*a*b^3 - 70*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x +
e)^2)*cos(f*x + e)/a) + (3*(3*a^4 - 33*a^3*b + 65*a^2*b^2 - 35*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 177*a^3*b + 4
45*a^2*b^2 - 315*a*b^3)*cos(f*x + e)^5 - (78*a^3*b - 305*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^3 - 5*(11*a^2*b^2 -
21*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^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)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(2*(tan((f*x+exp(1))/2)^2*(tan((f*x+
exp(1))/2)^2*(tan((f*x+exp(1))/2)^2*(tan((f*x+exp(1))/2)^2*(-tan((f*x+exp(1))/2)^2*(39582418599936*a^15*b^4-79
164837199872*a^16*b^3+39582418599936*a^17*b^2)/(-10133099161583616*a^16*b^4*sign(tan((f*x+exp(1))/2)^2-1)+2026
6198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1)-10133099161583616*a^18*b^2*sign(tan((f*x+exp(1))/2)^2-1))
-(-554153860399104*a^14*b^5+1306219813797888*a^15*b^4-949978046398464*a^16*b^3+197912092999680*a^17*b^2)/(-101
33099161583616*a^16*b^4*sign(tan((f*x+exp(1))/2)^2-1)+20266198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1)
-10133099161583616*a^18*b^2*sign(tan((f*x+exp(1))/2)^2-1)))-(-14777436277309440*a^13*b^6+40321290413801472*a^1
4*b^5-37497744553672704*a^15*b^4+13141362975178752*a^16*b^3-1187472557998080*a^17*b^2)/(-10133099161583616*a^1
6*b^4*sign(tan((f*x+exp(1))/2)^2-1)+20266198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1)-10133099161583616
*a^18*b^2*sign(tan((f*x+exp(1))/2)^2-1)))-(-44332308831928320*a^12*b^7+133630245193383936*a^13*b^6-14993820165
6557568*a^14*b^5+78294023990673408*a^15*b^4-19632879625568256*a^16*b^3+1979120929996800*a^17*b^2)/(-1013309916
1583616*a^16*b^4*sign(tan((f*x+exp(1))/2)^2-1)+20266198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1)-101330
99161583616*a^18*b^2*sign(tan((f*x+exp(1))/2)^2-1)))-(-30399297484750848*a^12*b^7+79164837199872000*a^13*b^6-6
5865144550293504*a^14*b^5+14447582788976640*a^15*b^4+4037406697193472*a^16*b^3-1385384650997760*a^17*b^2)/(-10
133099161583616*a^16*b^4*sign(tan((f*x+exp(1))/2)^2-1)+20266198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1
)-10133099161583616*a^18*b^2*sign(tan((f*x+exp(1))/2)^2-1)))-(-8444249301319680*a^13*b^6+21084234974232576*a^1
4*b^5-16479480277106688*a^15*b^4+3483252836794368*a^16*b^3+356241767399424*a^17*b^2)/(-10133099161583616*a^16*
b^4*sign(tan((f*x+exp(1))/2)^2-1)+20266198323167232*a^17*b^3*sign(tan((f*x+exp(1))/2)^2-1)-10133099161583616*a
^18*b^2*sign(tan((f*x+exp(1))/2)^2-1)))/sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+ex
p(1))/2)^2+a)/(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a)+2*(-1/64*(-6*a^3
*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2
)^2+a))-26*a*b^2*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*ta
n((f*x+exp(1))/2)^2+a))+24*a^2*b*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp
(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))+4*a^2*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a
*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))^3+22*b^2*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+
exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))^3-20*a*b*(-sqrt(a)*tan((f*x+exp(1))/2)^2+
sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))^3+3*sqrt(a)*a^2*(-sqrt(a)
*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))^2-
8*sqrt(a)*a*b*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((
f*x+exp(1))/2)^2+a))^2-5*sqrt(a)*a^3+12*sqrt(a)*a^2*b)/a^4/(-(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+e
xp(1))/2)^4-2*a*tan((f*x+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))^2+a)^2/sign(tan((f*x+exp(1))/2)^2-1)-1/32*
(3*a^2+35*b^2-30*a*b)*atan((-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x+exp(1))/2
)^2+4*b*tan((f*x+exp(1))/2)^2+a))/sqrt(-a))/a^4/sqrt(-a)/sign(tan((f*x+exp(1))/2)^2-1)-1/64*(3*sqrt(a)*a^2+35*
sqrt(a)*b^2-30*sqrt(a)*a*b)*ln(abs(a*(-sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a*tan((f*x+exp(1))/2)^4-2*a*tan((f*x
+exp(1))/2)^2+4*b*tan((f*x+exp(1))/2)^2+a))+sqrt(a)*a-2*sqrt(a)*b))/a^5/sign(tan((f*x+exp(1))/2)^2-1)))

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maple [B]  time = 7.12, size = 49917, normalized size = 210.62 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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