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*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)

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Rubi [A]  time = 0.366037, antiderivative size = 257, normalized size of antiderivative = 1., 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 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
]

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 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 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 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 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 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]

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*}

Mathematica [C]  time = 5.23082, size = 549, normalized size = 2.14 \[ \frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-48 \left (a^2-10 a b+5 b^2\right ) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right ) (a \cos (2 (e+f x))+a+2 b)^2+48 \left (a^2-10 a b+5 b^2\right ) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right ) (a \cos (2 (e+f x))+a+2 b)^2+\frac{48 \sqrt{b} \left (5 a^2-10 a b+b^2\right ) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )}{\sqrt{a}}+\frac{48 \sqrt{b} \left (5 a^2-10 a b+b^2\right ) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )}{\sqrt{a}}-2 (a+b) \cot (e+f x) \csc ^3(e+f x) \left (\left (78 a^2 b+35 a^3-93 a b^2+224 b^3\right ) \cos (2 (e+f x))+2 \left (-8 a^2 b+a^3+53 a b^2-10 b^3\right ) \cos (4 (e+f x))+18 a^2 b \cos (6 (e+f x))+112 a^2 b-3 a^3 \cos (6 (e+f x))+30 a^3-3 a b^2 \cos (6 (e+f x))+182 a b^2-140 b^3\right )\right )}{1024 f (a+b)^5 \left (a+b \sec ^2(e+f x)\right )^3} \]

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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Maple [B]  time = 0.123, size = 567, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

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]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.36488, size = 4163, normalized size = 16.2 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.39203, size = 1889, normalized size = 7.35 \begin{align*} \text{result too large to display} \end{align*}

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]

1/64*(12*(a^2 - 10*a*b + 5*b^2)*log(-(cos(f*x + e) - 1)/(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) - 24*(5*a^2*b - 10*a*b^2 + b^3)*arctan(-(a*cos(f*x + e) - b)/(sqrt(a*b)*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)) - (8*a^3*(cos(f*x + e) - 1)
/(cos(f*x + e) + 1) - 24*a*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 16*b^3*(cos(f*x + e) - 1)/(cos(f*x + e)
+ 1) - a^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 3*a^2*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 3*
a*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/(a^6 + 6*a^5*
b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6) - (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*a^4
*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 24*a^2*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 32*a*b^3*(cos(f*x
+ e) - 1)/(cos(f*x + e) + 1) + 12*b^4*(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 + 136*a^3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 224*a^2*b^2*(cos(f*x + e) - 1)^2/(cos
(f*x + e) + 1)^2 - 40*a*b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 108*b^4*(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 + 280*a^3*b*(cos(f*x + e) - 1)^3/(cos(f*x + e)
+ 1)^3 - 64*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 152*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) +
1)^3 + 212*b^4*(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 +
84*a^3*b*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 + 30*a^2*b^2*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 + 84
*a*b^3*(cos(f*x + e) - 1)^4/(cos(f*x + e) + 1)^4 - 123*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 - 104*a^3*b*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 24*a^2*b^2*(
cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 + 72*a*b^3*(cos(f*x + e) - 1)^5/(cos(f*x + e) + 1)^5 - 24*b^4*(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 - 48*a^3*b*(cos(f*x + e)
- 1)^6/(cos(f*x + e) + 1)^6 - 84*a^2*b^2*(cos(f*x + e) - 1)^6/(cos(f*x + e) + 1)^6 + 30*b^4*(cos(f*x + e) - 1)
^6/(cos(f*x + e) + 1)^6)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*(a*(cos(f*x + e) - 1)/(cos
(f*x + e) + 1) + b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 2*a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 2*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 + b*(cos(f*x + e) - 1
)^3/(cos(f*x + e) + 1)^3)^2))/f