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

Optimal. Leaf size=259 \[ -\frac{3 \left (a^2-12 a b+16 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac{3 \sqrt{b} \left (5 a^2-20 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right )}{8 a^5 f \sqrt{a-b}}-\frac{3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac{b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \]

[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))

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Rubi [A]  time = 0.376137, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3664, 470, 527, 522, 207, 205} \[ -\frac{3 \left (a^2-12 a b+16 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac{3 \sqrt{b} \left (5 a^2-20 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right )}{8 a^5 f \sqrt{a-b}}-\frac{3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac{b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \]

Antiderivative was successfully verified.

[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 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]

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 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 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 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 \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \tan ^{-1}\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 ) \tanh ^{-1}(\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]  time = 6.56108, size = 468, normalized size = 1.81 \[ -\frac{3 \left (3 a b \cos (e+f x)-4 b^2 \cos (e+f x)\right )}{4 a^4 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}+\frac{b^2 \cos (e+f x)}{a^3 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)^2}+\frac{3 \left (a^2-12 a b+16 b^2\right ) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{8 a^5 f}-\frac{3 \left (a^2-12 a b+16 b^2\right ) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{8 a^5 f}-\frac{3 \sqrt{b} \sqrt{a-b} \left (5 a^2-20 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (e+f x)\right ) \left (\sqrt{a-b} \cos \left (\frac{1}{2} (e+f x)\right )-\sqrt{a} \sin \left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{b}}\right )}{8 a^5 f (b-a)}-\frac{3 \sqrt{b} \sqrt{a-b} \left (5 a^2-20 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (e+f x)\right ) \left (\sqrt{a-b} \cos \left (\frac{1}{2} (e+f x)\right )+\sqrt{a} \sin \left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{b}}\right )}{8 a^5 f (b-a)}-\frac{3 (a-4 b) \csc ^2\left (\frac{1}{2} (e+f x)\right )}{32 a^4 f}+\frac{3 (a-4 b) \sec ^2\left (\frac{1}{2} (e+f x)\right )}{32 a^4 f}-\frac{\csc ^4\left (\frac{1}{2} (e+f x)\right )}{64 a^3 f}+\frac{\sec ^4\left (\frac{1}{2} (e+f x)\right )}{64 a^3 f} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [B]  time = 0.112, size = 560, 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*tan(f*x+e)^2)^3,x)

[Out]

1/16/f/a^3/(cos(f*x+e)+1)^2+3/16/f/a^3/(cos(f*x+e)+1)-3/4/f/a^4/(cos(f*x+e)+1)*b-3/16/f/a^3*ln(cos(f*x+e)+1)+9
/4/f/a^4*ln(cos(f*x+e)+1)*b-3/f/a^5*ln(cos(f*x+e)+1)*b^2-9/8/f*b/a^2/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*cos(f
*x+e)^3+21/8/f*b^2/a^3/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*cos(f*x+e)^3-3/2/f*b^3/a^4/(a*cos(f*x+e)^2-cos(f*x+
e)^2*b+b)^2*cos(f*x+e)^3-7/8/f*b^2/a^3/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*cos(f*x+e)+3/2/f*b^3/a^4/(a*cos(f*x
+e)^2-cos(f*x+e)^2*b+b)^2*cos(f*x+e)+15/8/f*b/a^3/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2))-15/
2/f*b^2/a^4/(b*(a-b))^(1/2)*arctan((a-b)*cos(f*x+e)/(b*(a-b))^(1/2))+6/f*b^3/a^5/(b*(a-b))^(1/2)*arctan((a-b)*
cos(f*x+e)/(b*(a-b))^(1/2))-1/16/f/a^3/(cos(f*x+e)-1)^2+3/16/f/a^3/(cos(f*x+e)-1)-3/4/f/a^4/(cos(f*x+e)-1)*b+3
/16/f/a^3*ln(cos(f*x+e)-1)-9/4/f/a^4*ln(cos(f*x+e)-1)*b+3/f/a^5*ln(cos(f*x+e)-1)*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*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.85351, size = 3969, normalized size = 15.32 \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*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)]

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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*tan(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.66481, size = 1241, normalized size = 4.79 \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*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/64*(12*(a^2 - 12*a*b + 16*b^2)*log(-(cos(f*x + e) - 1)/(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)))/(sqrt(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*(c
os(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*(co
s(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)^2))/f