3.59 \(\int \frac {\csc ^5(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=257 \[ \frac {3 \left (a^2-6 a b+b^2\right ) \cos (e+f x)}{8 f (a+b)^4 \left (a \cos ^2(e+f x)+b\right )}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{8 f (a+b)^3 \left (a \cos ^2(e+f x)+b\right )^2}+\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 \sqrt {a} f (a+b)^5}-\frac {3 \left (a^2-10 a b+5 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^5}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {(a-7 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left (a \cos ^2(e+f x)+b\right )^2} \]

[Out]

-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)*cos(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+3/8*(5*a^2-10*a*b+b^2)*arctan(
cos(f*x+e)*a^(1/2)/b^(1/2))*b^(1/2)/(a+b)^5/f/a^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4133, 470, 578, 527, 522, 206, 205} \[ \frac {3 \left (a^2-6 a b+b^2\right ) \cos (e+f x)}{8 f (a+b)^4 \left (a \cos ^2(e+f x)+b\right )}+\frac {\left (a^2-9 a b+2 b^2\right ) \cos (e+f x)}{8 f (a+b)^3 \left (a \cos ^2(e+f x)+b\right )^2}+\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 \sqrt {a} f (a+b)^5}-\frac {3 \left (a^2-10 a b+5 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^5}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {(a-7 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left (a \cos ^2(e+f x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*Sqrt[a]*(a + b)^5*f) - (3*(a^2 -
10*a*b + 5*b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^5*f) + ((a^2 - 9*a*b + 2*b^2)*Cos[e + f*x])/(8*(a + b)^3*f*(
b + a*Cos[e + f*x]^2)^2) + (3*(a^2 - 6*a*b + b^2)*Cos[e + f*x])/(8*(a + b)^4*f*(b + a*Cos[e + f*x]^2)) - ((a -
7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(
a + b)*f*(b + a*Cos[e + f*x]^2)^2)

Rule 205

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

Rule 206

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

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 522

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 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 578

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c -
a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 4133

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*} \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3 \left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 b+(-a+4 b) x^2\right )}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{4 (a+b) f}\\ &=-\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}+\frac {\operatorname {Subst}\left (\int \frac {(a-7 b) b+\left (-3 a^2+29 a b-8 b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{8 (a+b)^2 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 {(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}-\frac {\operatorname {Subst}\left (\int \frac {-12 (a-3 b) b^2+12 b \left (a^2-9 a b+2 b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{32 b (a+b)^3 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}+\frac {\operatorname {Subst}\left (\int \frac {96 (a-b) b^3-24 b^2 \left (a^2-6 a b+b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cos (e+f x)\right )}{64 b^2 (a+b)^4 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}+\frac {\left (3 b \left (5 a^2-10 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 (a+b)^5 f}-\frac {\left (3 \left (a^2-10 a b+5 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (e+f x)\right )}{8 (a+b)^5 f}\\ &=\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\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 ) \tanh ^{-1}(\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}\\ \end {align*}

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Mathematica [C]  time = 5.10, size = 549, normalized size = 2.14 \[ \frac {\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-48 \left (a^2-10 a b+5 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (2 (e+f x))+a+2 b)^2+48 \left (a^2-10 a b+5 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (2 (e+f x))+a+2 b)^2+\frac {48 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {48 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}-2 (a+b) \cot (e+f x) \csc ^3(e+f x) \left (-3 a^3 \cos (6 (e+f x))+30 a^3+18 a^2 b \cos (6 (e+f x))+112 a^2 b+\left (35 a^3+78 a^2 b-93 a b^2+224 b^3\right ) \cos (2 (e+f x))+2 \left (a^3-8 a^2 b+53 a b^2-10 b^3\right ) \cos (4 (e+f x))-3 a b^2 \cos (6 (e+f x))+182 a b^2-140 b^3\right )\right )}{1024 f (a+b)^5 \left (a+b \sec ^2(e+f x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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[(Co
s[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)/Sqrt[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)/(102
4*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)

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fricas [B]  time = 1.00, size = 1833, normalized size = 7.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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 - 26*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)*sqrt(-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 + 4
6*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))/((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)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*
b^5 + 5*a*b^6 + b^7)*f), 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 - 26*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 + 6*((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)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/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)*cos(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)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)]

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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*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)2/f*((32*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b^3+96*((
1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b^2*a+96*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b*a^2+32*((1-co
s(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*a^3-512*(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*b^3-768*(1-cos(f*x+exp(1
)))/(1+cos(f*x+exp(1)))*b^2*a+256*(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*a^3)/(4096*b^6+24576*b^5*a+61440*b^4
*a^2+81920*b^3*a^3+61440*b^2*a^4+24576*b*a^5+4096*a^6)+(-30*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^6*b^4+84
*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^6*b^2*a^2+48*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^6*b*a^3-6*((
1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^6*a^4-24*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^5*b^4+72*((1-cos(f*
x+exp(1)))/(1+cos(f*x+exp(1))))^5*b^3*a-24*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^5*b^2*a^2-104*((1-cos(f*x
+exp(1)))/(1+cos(f*x+exp(1))))^5*b*a^3+16*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^5*a^4+123*((1-cos(f*x+exp(
1)))/(1+cos(f*x+exp(1))))^4*b^4-84*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^4*b^3*a-30*((1-cos(f*x+exp(1)))/(
1+cos(f*x+exp(1))))^4*b^2*a^2-84*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^4*b*a^3-5*((1-cos(f*x+exp(1)))/(1+c
os(f*x+exp(1))))^4*a^4+212*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^3*b^4-152*((1-cos(f*x+exp(1)))/(1+cos(f*x
+exp(1))))^3*b^3*a-64*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^3*b^2*a^2+280*((1-cos(f*x+exp(1)))/(1+cos(f*x+
exp(1))))^3*b*a^3-20*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^3*a^4+108*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1
))))^2*b^4+40*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b^3*a-224*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^
2*b^2*a^2-136*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b*a^3+20*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2
*a^4+12*(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*b^4+32*(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*b^3*a+24*(1-cos
(f*x+exp(1)))/(1+cos(f*x+exp(1)))*b^2*a^2-4*(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*a^4-b^4-4*b^3*a-6*b^2*a^2-
4*b*a^3-a^4)/(128*b^5+640*b^4*a+1280*b^3*a^2+1280*b^2*a^3+640*b*a^4+128*a^5)/(((1-cos(f*x+exp(1)))/(1+cos(f*x+
exp(1))))^3*b+((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^3*a+2*((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*b-2*
((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1))))^2*a+(1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))*b+(1-cos(f*x+exp(1)))/(1
+cos(f*x+exp(1)))*a)^2+(15*b^2-30*b*a+3*a^2)/(32*b^5+160*b^4*a+320*b^3*a^2+320*b^2*a^3+160*b*a^4+32*a^5)*ln(ab
s(1-cos(f*x+exp(1)))/abs(1+cos(f*x+exp(1))))+(-3*b^3+30*b^2*a-15*b*a^2)*1/4/(4*b^5+20*b^4*a+40*b^3*a^2+40*b^2*
a^3+20*b*a^4+4*a^5)/sqrt(a*b)*atan((-a*cos(f*x+exp(1))+b)/(sqrt(a*b)*cos(f*x+exp(1))+sqrt(a*b))))

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maple [B]  time = 1.24, size = 567, normalized size = 2.21 \[ -\frac {9 b \left (\cos ^{3}\left (f x +e \right )\right ) a^{3}}{8 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {3 b^{2} \left (\cos ^{3}\left (f x +e \right )\right ) a^{2}}{4 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 b^{3} \left (\cos ^{3}\left (f x +e \right )\right ) a}{8 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {7 b^{2} \cos \left (f x +e \right ) a^{2}}{8 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {b^{3} \cos \left (f x +e \right ) a}{4 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {5 b^{4} \cos \left (f x +e \right )}{8 f \left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {15 b \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right ) a^{2}}{8 f \left (a +b \right )^{5} \sqrt {a b}}-\frac {15 b^{2} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right ) a}{4 f \left (a +b \right )^{5} \sqrt {a b}}+\frac {3 b^{3} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 f \left (a +b \right )^{5} \sqrt {a b}}-\frac {1}{16 f \left (a +b \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{2}}+\frac {3 a}{16 f \left (a +b \right )^{4} \left (-1+\cos \left (f x +e \right )\right )}-\frac {9 b}{16 f \left (a +b \right )^{4} \left (-1+\cos \left (f x +e \right )\right )}+\frac {3 \ln \left (-1+\cos \left (f x +e \right )\right ) a^{2}}{16 f \left (a +b \right )^{5}}-\frac {15 \ln \left (-1+\cos \left (f x +e \right )\right ) a b}{8 f \left (a +b \right )^{5}}+\frac {15 \ln \left (-1+\cos \left (f x +e \right )\right ) b^{2}}{16 f \left (a +b \right )^{5}}+\frac {1}{16 f \left (a +b \right )^{3} \left (1+\cos \left (f x +e \right )\right )^{2}}+\frac {3 a}{16 f \left (a +b \right )^{4} \left (1+\cos \left (f x +e \right )\right )}-\frac {9 b}{16 f \left (a +b \right )^{4} \left (1+\cos \left (f x +e \right )\right )}-\frac {3 \ln \left (1+\cos \left (f x +e \right )\right ) a^{2}}{16 f \left (a +b \right )^{5}}+\frac {15 \ln \left (1+\cos \left (f x +e \right )\right ) a b}{8 f \left (a +b \right )^{5}}-\frac {15 \ln \left (1+\cos \left (f x +e \right )\right ) b^{2}}{16 f \left (a +b \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x)

[Out]

-9/8/f*b/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)^3*a^3-3/4/f*b^2/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)^3*a^2
+3/8/f*b^3/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)^3*a-7/8/f*b^2/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)*a^2-1
/4/f*b^3/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)*a+5/8/f*b^4/(a+b)^5/(b+a*cos(f*x+e)^2)^2*cos(f*x+e)+15/8/f*b/
(a+b)^5/(a*b)^(1/2)*arctan(a*cos(f*x+e)/(a*b)^(1/2))*a^2-15/4/f*b^2/(a+b)^5/(a*b)^(1/2)*arctan(a*cos(f*x+e)/(a
*b)^(1/2))*a+3/8/f*b^3/(a+b)^5/(a*b)^(1/2)*arctan(a*cos(f*x+e)/(a*b)^(1/2))-1/16/f/(a+b)^3/(-1+cos(f*x+e))^2+3
/16/f/(a+b)^4/(-1+cos(f*x+e))*a-9/16/f/(a+b)^4/(-1+cos(f*x+e))*b+3/16/f/(a+b)^5*ln(-1+cos(f*x+e))*a^2-15/8/f/(
a+b)^5*ln(-1+cos(f*x+e))*a*b+15/16/f/(a+b)^5*ln(-1+cos(f*x+e))*b^2+1/16/f/(a+b)^3/(1+cos(f*x+e))^2+3/16/f/(a+b
)^4/(1+cos(f*x+e))*a-9/16/f/(a+b)^4/(1+cos(f*x+e))*b-3/16/f/(a+b)^5*ln(1+cos(f*x+e))*a^2+15/8/f/(a+b)^5*ln(1+c
os(f*x+e))*a*b-15/16/f/(a+b)^5*ln(1+cos(f*x+e))*b^2

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maxima [B]  time = 0.44, size = 528, normalized size = 2.05 \[ -\frac {\frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {3 \, {\left (a^{2} - 10 \, a b + 5 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {6 \, {\left (5 \, a^{2} b - 10 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \sqrt {a b}} - \frac {2 \, {\left (3 \, {\left (a^{3} - 6 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{3} - 31 \, a^{2} b + 31 \, a b^{2} - 5 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{2} b - 34 \, a b^{2} + 19 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )}}{{\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (f x + e\right )^{8} - 2 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (f x + e\right )^{6} + a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6} + {\left (a^{6} - 9 \, a^{4} b^{2} - 16 \, a^{3} b^{3} - 9 \, a^{2} b^{4} + b^{6}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b + 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} - 3 \, a b^{5} - b^{6}\right )} \cos \left (f x + e\right )^{2}}}{16 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*c
os(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)*cos(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

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mupad [B]  time = 9.60, size = 5613, normalized size = 21.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 + 10752*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 + 10*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 - 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 + 10752*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)))/(((135*a*b^7)/256 + (135*a^7*b)/256 - (1215*a^2*b^6)/128 + (13257*a^3*b^5)/256 - (5913
*a^4*b^4)/64 + (13257*a^5*b^3)/256 - (1215*a^6*b^2)/128)/(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)*(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 + 25
2*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 - 23
04*a^3*b^10 - 8960*a^4*b^9 - 19200*a^5*b^8 - 23040*a^6*b^7 - 10752*a^7*b^6 + 10752*a^8*b^5 + 23040*a^9*b^4 + 1
9200*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 + 10*a^3*b^3 + 10*a^4*b^2)))*(-a*b)^(1/2)*(5*a^2 - 10*a*b + b^2))/(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)) + (3*((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^1
1 - 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 - 23
040*a^6*b^7 - 10752*a^7*b^6 + 10752*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 + 10*a
^3*b^3 + 10*a^4*b^2)))*(-a*b)^(1/2)*(5*a^2 - 10*a*b + b^2))/(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))))*(-a*b)^(1/2)*(5*a^2 - 10*a*b + b^2)*3i)/(8*f*(a*b^5 + 5*a^5*b + a^6 + 5*a^2*b^4 + 10*a^3*b^
3 + 10*a^4*b^2)) - (atan(((((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) - (cos(e + f*x)*(3/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*(230
4*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 + 10752*a^8*b^5 + 23040*a^9*b^4 + 19200*a^10*b^3 + 8960*a^11*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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b
^2)/(a + b)^5) + (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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*1i - (((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) + (cos(e + f*x)*(3/(16*(a + b)^3) - (
9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*(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 + 10752*a^8*b^5 + 23040*a^9*b^4 + 19200*a^10*b^3 + 8960*a^11*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
/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5) - (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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*1
i)/((((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^1
0*b^2) - (cos(e + f*x)*(3/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*(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 + 10752*a^8*b^5 +
23040*a^9*b^4 + 19200*a^10*b^3 + 8960*a^11*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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5) + (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/(16*(a + b)^3)
- (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5) - ((135*a*b^7)/256 + (135*a^7*b)/256 - (1215*a^2*b^6)/128 + (13257*
a^3*b^5)/256 - (5913*a^4*b^4)/64 + (13257*a^5*b^3)/256 - (1215*a^6*b^2)/128)/(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) + (((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^1
2 + 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 + 2
20*a^9*b^3 + 66*a^10*b^2) + (cos(e + f*x)*(3/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)*(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
+ 10752*a^8*b^5 + 23040*a^9*b^4 + 19200*a^10*b^3 + 8960*a^11*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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)
/(a + b)^5) - (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/(16*(a + b)^3) - (9*b)/(4*(a + b)^4) + (3*b^2)/(a + b)^5)))*(3i/(8*(a + b)^3) - (b*9i)/(2*(a + b)^4) + (b^
2*6i)/(a + b)^5))/f - ((cos(e + f*x)^3*(19*a^2*b - 34*a*b^2 + 19*b^3))/(8*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a
^2*b^2)) + (cos(e + f*x)^5*(31*a*b^2 - 31*a^2*b + 5*a^3 - 5*b^3))/(8*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^
2)) + (3*b^2*cos(e + f*x)*(a - b))/(2*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)) - (3*a*cos(e + f*x)^7*(a^2
- 6*a*b + b^2))/(8*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)))/(f*(cos(e + f*x)^4*(a^2 - 4*a*b + b^2) + b^2
+ cos(e + f*x)^6*(2*a*b - 2*a^2) + cos(e + f*x)^2*(2*a*b - 2*b^2) + a^2*cos(e + f*x)^8))

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sympy [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*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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