3.361 \(\int \frac {\cot ^4(e+f x)}{(a+b \sec ^2(e+f x))^2} \, dx\)
Optimal. Leaf size=160 \[ -\frac {b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 f (a+b)^{7/2}}+\frac {\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a f (a+b)^3}+\frac {x}{a^2}-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac {b \cot ^3(e+f x)}{2 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
[Out]
x/a^2-1/2*b^(5/2)*(7*a+2*b)*arctan(b^(1/2)*tan(f*x+e)/(a+b)^(1/2))/a^2/(a+b)^(7/2)/f+1/2*(2*a^2+6*a*b-b^2)*cot
(f*x+e)/a/(a+b)^3/f-1/6*(2*a-3*b)*cot(f*x+e)^3/a/(a+b)^2/f-1/2*b*cot(f*x+e)^3/a/(a+b)/f/(a+b+b*tan(f*x+e)^2)
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Rubi [A] time = 0.35, antiderivative size = 160, 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 =
{4141, 1975, 472, 583, 522, 203, 205} \[ -\frac {b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 f (a+b)^{7/2}}+\frac {\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a f (a+b)^3}+\frac {x}{a^2}-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac {b \cot ^3(e+f x)}{2 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]
[Out]
x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*f) + ((2*a^2 + 6
*a*b - b^2)*Cot[e + f*x])/(2*a*(a + b)^3*f) - ((2*a - 3*b)*Cot[e + f*x]^3)/(6*a*(a + b)^2*f) - (b*Cot[e + f*x]
^3)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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]
Rule 472
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && 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 583
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 1975
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&& !BinomialMatchQ[{u, v}, x]
Rule 4141
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 a-3 b-5 b x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a (a+b) f}\\ &=-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac {b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {3 \left (2 a^2+6 a b-b^2\right )+3 (2 a-3 b) b x^2}{x^2 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a (a+b)^2 f}\\ &=\frac {\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac {b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {3 \left (2 a^3+8 a^2 b+12 a b^2+b^3\right )+3 b \left (2 a^2+6 a b-b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a (a+b)^3 f}\\ &=\frac {\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac {b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f}-\frac {\left (b^3 (7 a+2 b)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 (a+b)^3 f}\\ &=\frac {x}{a^2}-\frac {b^{5/2} (7 a+2 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{7/2} f}+\frac {\left (2 a^2+6 a b-b^2\right ) \cot (e+f x)}{2 a (a+b)^3 f}-\frac {(2 a-3 b) \cot ^3(e+f x)}{6 a (a+b)^2 f}-\frac {b \cot ^3(e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 6.80, size = 1588, normalized size = 9.92 \[ \frac {(\cos (2 (e+f x)) a+a+2 b) \sec ^4(e+f x) \left (\frac {48 (7 a+2 b) \tan ^{-1}\left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 (e+f x)) a+a+2 b) (\cos (2 e)-i \sin (2 e)) b^3}{\sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}+\csc (e) \csc ^3(e+f x) \sec (2 e) \left (6 f x \cos (2 e-f x) a^4+6 f x \cos (2 e+f x) a^4-6 f x \cos (4 e+f x) a^4-3 f x \cos (2 e+3 f x) a^4+3 f x \cos (4 e+3 f x) a^4-3 f x \cos (6 e+3 f x) a^4-3 f x \cos (2 e+5 f x) a^4+3 f x \cos (4 e+5 f x) a^4-3 f x \cos (6 e+5 f x) a^4+3 f x \cos (8 e+5 f x) a^4-12 \sin (f x) a^4+4 \sin (3 f x) a^4+4 \sin (2 e-f x) a^4-4 \sin (2 e+f x) a^4-12 \sin (4 e+f x) a^4-12 \sin (2 e+3 f x) a^4+4 \sin (4 e+3 f x) a^4-12 \sin (6 e+3 f x) a^4+8 \sin (2 e+5 f x) a^4+8 \sin (6 e+5 f x) a^4+54 b f x \cos (2 e-f x) a^3+54 b f x \cos (2 e+f x) a^3-54 b f x \cos (4 e+f x) a^3+3 b f x \cos (2 e+3 f x) a^3-3 b f x \cos (4 e+3 f x) a^3+3 b f x \cos (6 e+3 f x) a^3-9 b f x \cos (2 e+5 f x) a^3+9 b f x \cos (4 e+5 f x) a^3-9 b f x \cos (6 e+5 f x) a^3+9 b f x \cos (8 e+5 f x) a^3-60 b \sin (f x) a^3+36 b \sin (3 f x) a^3+76 b \sin (2 e-f x) a^3-76 b \sin (2 e+f x) a^3-60 b \sin (4 e+f x) a^3-24 b \sin (2 e+3 f x) a^3+36 b \sin (4 e+3 f x) a^3-24 b \sin (6 e+3 f x) a^3+20 b \sin (2 e+5 f x) a^3+20 b \sin (6 e+5 f x) a^3+126 b^2 f x \cos (2 e-f x) a^2+126 b^2 f x \cos (2 e+f x) a^2-126 b^2 f x \cos (4 e+f x) a^2+27 b^2 f x \cos (2 e+3 f x) a^2-27 b^2 f x \cos (4 e+3 f x) a^2+27 b^2 f x \cos (6 e+3 f x) a^2-9 b^2 f x \cos (2 e+5 f x) a^2+9 b^2 f x \cos (4 e+5 f x) a^2-9 b^2 f x \cos (6 e+5 f x) a^2+9 b^2 f x \cos (8 e+5 f x) a^2-96 b^2 \sin (f x) a^2+80 b^2 \sin (3 f x) a^2+144 b^2 \sin (2 e-f x) a^2-144 b^2 \sin (2 e+f x) a^2-96 b^2 \sin (4 e+f x) a^2+80 b^2 \sin (4 e+3 f x) a^2+114 b^3 f x \cos (2 e-f x) a+114 b^3 f x \cos (2 e+f x) a-114 b^3 f x \cos (4 e+f x) a+33 b^3 f x \cos (2 e+3 f x) a-33 b^3 f x \cos (4 e+3 f x) a+33 b^3 f x \cos (6 e+3 f x) a-3 b^3 f x \cos (2 e+5 f x) a+3 b^3 f x \cos (4 e+5 f x) a-3 b^3 f x \cos (6 e+5 f x) a+3 b^3 f x \cos (8 e+5 f x) a-6 b^3 \sin (3 f x) a+6 b^3 \sin (2 e+f x) a-6 b^3 \sin (4 e+f x) a+6 b^3 \sin (2 e+3 f x) a-3 b^3 \sin (4 e+3 f x) a+3 b^3 \sin (6 e+3 f x) a+3 b^3 \sin (2 e+5 f x) a-3 b^3 \sin (4 e+5 f x) a-6 (a+b)^3 (a+6 b) f x \cos (f x)+3 (a-4 b) (a+b)^3 f x \cos (3 f x)+36 b^4 f x \cos (2 e-f x)+36 b^4 f x \cos (2 e+f x)-36 b^4 f x \cos (4 e+f x)+12 b^4 f x \cos (2 e+3 f x)-12 b^4 f x \cos (4 e+3 f x)+12 b^4 f x \cos (6 e+3 f x)+18 b^4 \sin (f x)+6 b^4 \sin (3 f x)+18 b^4 \sin (2 e-f x)+18 b^4 \sin (2 e+f x)-18 b^4 \sin (4 e+f x)-6 b^4 \sin (2 e+3 f x)-6 b^4 \sin (4 e+3 f x)+6 b^4 \sin (6 e+3 f x)\right )\right )}{384 a^2 (a+b)^3 f \left (b \sec ^2(e+f x)+a\right )^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]
[Out]
((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((48*b^3*(7*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(
-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(
e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + Csc[e]*Csc[e + f*x]^3*Sec[2*
e]*(-6*(a + b)^3*(a + 6*b)*f*x*Cos[f*x] + 3*(a - 4*b)*(a + b)^3*f*x*Cos[3*f*x] + 6*a^4*f*x*Cos[2*e - f*x] + 54
*a^3*b*f*x*Cos[2*e - f*x] + 126*a^2*b^2*f*x*Cos[2*e - f*x] + 114*a*b^3*f*x*Cos[2*e - f*x] + 36*b^4*f*x*Cos[2*e
- f*x] + 6*a^4*f*x*Cos[2*e + f*x] + 54*a^3*b*f*x*Cos[2*e + f*x] + 126*a^2*b^2*f*x*Cos[2*e + f*x] + 114*a*b^3*
f*x*Cos[2*e + f*x] + 36*b^4*f*x*Cos[2*e + f*x] - 6*a^4*f*x*Cos[4*e + f*x] - 54*a^3*b*f*x*Cos[4*e + f*x] - 126*
a^2*b^2*f*x*Cos[4*e + f*x] - 114*a*b^3*f*x*Cos[4*e + f*x] - 36*b^4*f*x*Cos[4*e + f*x] - 3*a^4*f*x*Cos[2*e + 3*
f*x] + 3*a^3*b*f*x*Cos[2*e + 3*f*x] + 27*a^2*b^2*f*x*Cos[2*e + 3*f*x] + 33*a*b^3*f*x*Cos[2*e + 3*f*x] + 12*b^4
*f*x*Cos[2*e + 3*f*x] + 3*a^4*f*x*Cos[4*e + 3*f*x] - 3*a^3*b*f*x*Cos[4*e + 3*f*x] - 27*a^2*b^2*f*x*Cos[4*e + 3
*f*x] - 33*a*b^3*f*x*Cos[4*e + 3*f*x] - 12*b^4*f*x*Cos[4*e + 3*f*x] - 3*a^4*f*x*Cos[6*e + 3*f*x] + 3*a^3*b*f*x
*Cos[6*e + 3*f*x] + 27*a^2*b^2*f*x*Cos[6*e + 3*f*x] + 33*a*b^3*f*x*Cos[6*e + 3*f*x] + 12*b^4*f*x*Cos[6*e + 3*f
*x] - 3*a^4*f*x*Cos[2*e + 5*f*x] - 9*a^3*b*f*x*Cos[2*e + 5*f*x] - 9*a^2*b^2*f*x*Cos[2*e + 5*f*x] - 3*a*b^3*f*x
*Cos[2*e + 5*f*x] + 3*a^4*f*x*Cos[4*e + 5*f*x] + 9*a^3*b*f*x*Cos[4*e + 5*f*x] + 9*a^2*b^2*f*x*Cos[4*e + 5*f*x]
+ 3*a*b^3*f*x*Cos[4*e + 5*f*x] - 3*a^4*f*x*Cos[6*e + 5*f*x] - 9*a^3*b*f*x*Cos[6*e + 5*f*x] - 9*a^2*b^2*f*x*Co
s[6*e + 5*f*x] - 3*a*b^3*f*x*Cos[6*e + 5*f*x] + 3*a^4*f*x*Cos[8*e + 5*f*x] + 9*a^3*b*f*x*Cos[8*e + 5*f*x] + 9*
a^2*b^2*f*x*Cos[8*e + 5*f*x] + 3*a*b^3*f*x*Cos[8*e + 5*f*x] - 12*a^4*Sin[f*x] - 60*a^3*b*Sin[f*x] - 96*a^2*b^2
*Sin[f*x] + 18*b^4*Sin[f*x] + 4*a^4*Sin[3*f*x] + 36*a^3*b*Sin[3*f*x] + 80*a^2*b^2*Sin[3*f*x] - 6*a*b^3*Sin[3*f
*x] + 6*b^4*Sin[3*f*x] + 4*a^4*Sin[2*e - f*x] + 76*a^3*b*Sin[2*e - f*x] + 144*a^2*b^2*Sin[2*e - f*x] + 18*b^4*
Sin[2*e - f*x] - 4*a^4*Sin[2*e + f*x] - 76*a^3*b*Sin[2*e + f*x] - 144*a^2*b^2*Sin[2*e + f*x] + 6*a*b^3*Sin[2*e
+ f*x] + 18*b^4*Sin[2*e + f*x] - 12*a^4*Sin[4*e + f*x] - 60*a^3*b*Sin[4*e + f*x] - 96*a^2*b^2*Sin[4*e + f*x]
- 6*a*b^3*Sin[4*e + f*x] - 18*b^4*Sin[4*e + f*x] - 12*a^4*Sin[2*e + 3*f*x] - 24*a^3*b*Sin[2*e + 3*f*x] + 6*a*b
^3*Sin[2*e + 3*f*x] - 6*b^4*Sin[2*e + 3*f*x] + 4*a^4*Sin[4*e + 3*f*x] + 36*a^3*b*Sin[4*e + 3*f*x] + 80*a^2*b^2
*Sin[4*e + 3*f*x] - 3*a*b^3*Sin[4*e + 3*f*x] - 6*b^4*Sin[4*e + 3*f*x] - 12*a^4*Sin[6*e + 3*f*x] - 24*a^3*b*Sin
[6*e + 3*f*x] + 3*a*b^3*Sin[6*e + 3*f*x] + 6*b^4*Sin[6*e + 3*f*x] + 8*a^4*Sin[2*e + 5*f*x] + 20*a^3*b*Sin[2*e
+ 5*f*x] + 3*a*b^3*Sin[2*e + 5*f*x] - 3*a*b^3*Sin[4*e + 5*f*x] + 8*a^4*Sin[6*e + 5*f*x] + 20*a^3*b*Sin[6*e + 5
*f*x])))/(384*a^2*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)
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fricas [B] time = 0.61, size = 979, normalized size = 6.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")
[Out]
[1/24*(4*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 8*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)
^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)
*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a
*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^
4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 24*((a^4 + 3*a
^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3
*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x + e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 +
2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e)),
1/12*(2*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 4*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)
^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)
*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(
f*x + e) - 6*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 12*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x
+ e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x
+ e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(
f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e))]
________________________________________________________________________________________
giac [A] time = 0.50, size = 220, normalized size = 1.38 \[ -\frac {\frac {3 \, b^{3} \tan \left (f x + e\right )}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}} + \frac {3 \, {\left (7 \, a b^{3} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {a b + b^{2}}} - \frac {6 \, {\left (f x + e\right )}}{a^{2}} - \frac {2 \, {\left (3 \, a \tan \left (f x + e\right )^{2} + 9 \, b \tan \left (f x + e\right )^{2} - a - b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
[Out]
-1/6*(3*b^3*tan(f*x + e)/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*(b*tan(f*x + e)^2 + a + b)) + 3*(7*a*b^3 + 2*b^4
)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))/((a^5 + 3*a^4*b + 3*a^3*b^2 +
a^2*b^3)*sqrt(a*b + b^2)) - 6*(f*x + e)/a^2 - 2*(3*a*tan(f*x + e)^2 + 9*b*tan(f*x + e)^2 - a - b)/((a^3 + 3*a
^2*b + 3*a*b^2 + b^3)*tan(f*x + e)^3))/f
________________________________________________________________________________________
maple [A] time = 1.34, size = 186, normalized size = 1.16 \[ -\frac {b^{3} \tan \left (f x +e \right )}{2 f a \left (a +b \right )^{3} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}-\frac {7 b^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{2 f a \left (a +b \right )^{3} \sqrt {\left (a +b \right ) b}}-\frac {b^{4} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,a^{2} \left (a +b \right )^{3} \sqrt {\left (a +b \right ) b}}-\frac {1}{3 f \left (a +b \right )^{2} \tan \left (f x +e \right )^{3}}+\frac {a}{f \left (a +b \right )^{3} \tan \left (f x +e \right )}+\frac {3 b}{f \left (a +b \right )^{3} \tan \left (f x +e \right )}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x)
[Out]
-1/2/f*b^3/a/(a+b)^3*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)-7/2/f*b^3/a/(a+b)^3/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/(
(a+b)*b)^(1/2))-1/f*b^4/a^2/(a+b)^3/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))-1/3/f/(a+b)^2/tan(f*x
+e)^3+1/f/(a+b)^3/tan(f*x+e)*a+3/f/(a+b)^3/tan(f*x+e)*b+1/f/a^2*arctan(tan(f*x+e))
________________________________________________________________________________________
maxima [A] time = 0.45, size = 235, normalized size = 1.47 \[ -\frac {\frac {3 \, {\left (7 \, a b^{3} + 2 \, b^{4}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {3 \, {\left (2 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{4} - 2 \, a^{3} - 4 \, a^{2} b - 2 \, a b^{2} + 2 \, {\left (3 \, a^{3} + 11 \, a^{2} b + 8 \, a b^{2}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (f x + e\right )^{5} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (f x + e\right )^{3}} - \frac {6 \, {\left (f x + e\right )}}{a^{2}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")
[Out]
-1/6*(3*(7*a*b^3 + 2*b^4)*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt((
a + b)*b)) - (3*(2*a^2*b + 6*a*b^2 - b^3)*tan(f*x + e)^4 - 2*a^3 - 4*a^2*b - 2*a*b^2 + 2*(3*a^3 + 11*a^2*b + 8
*a*b^2)*tan(f*x + e)^2)/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(f*x + e)^5 + (a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*tan(f*x + e)^3) - 6*(f*x + e)/a^2)/f
________________________________________________________________________________________
mupad [B] time = 10.46, size = 4987, normalized size = 31.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e + f*x)^4/(a + b/cos(e + f*x)^2)^2,x)
[Out]
((tan(e + f*x)^2*(3*a + 8*b))/(3*(a + b)^2) - 1/(3*(a + b)) + (tan(e + f*x)^4*(6*a*b^2 + 2*a^2*b - b^3))/(2*a*
(a + b)^3))/(f*(tan(e + f*x)^3*(a + b) + b*tan(e + f*x)^5)) + atan((560*a^3*b^16*tan(e + f*x))/(560*a^3*b^16 +
7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 6998
40*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16
*b^3 + 64*a^17*b^2) + (7280*a^4*b^15*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6
*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*
b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (42560*a^5*b^14*tan(e
+ f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 +
741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 76
80*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (149184*a^6*b^13*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42
560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 50500
8*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2)
+ (351904*a^7*b^12*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^
7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13
*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (593440*a^8*b^11*tan(e + f*x))/(560*a^3
*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10
+ 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 10
24*a^16*b^3 + 64*a^17*b^2) + (741120*a^9*b^10*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 1
49184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278
768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (699840*a^10*
b^9*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*
a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^1
4*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (505008*a^11*b^8*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4
*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b
^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64
*a^17*b^2) + (278768*a^12*b^7*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 +
351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 1
16480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (116480*a^13*b^6*tan(e + f*x)
)/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 74112
0*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^1
5*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (35840*a^14*b^5*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5
*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*
b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (768
0*a^15*b^4*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 +
593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35
840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2) + (1024*a^16*b^3*tan(e + f*x))/(560*a^3*b^16 + 728
0*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13 + 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a
^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 + 116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3
+ 64*a^17*b^2) + (64*a^17*b^2*tan(e + f*x))/(560*a^3*b^16 + 7280*a^4*b^15 + 42560*a^5*b^14 + 149184*a^6*b^13
+ 351904*a^7*b^12 + 593440*a^8*b^11 + 741120*a^9*b^10 + 699840*a^10*b^9 + 505008*a^11*b^8 + 278768*a^12*b^7 +
116480*a^13*b^6 + 35840*a^14*b^5 + 7680*a^15*b^4 + 1024*a^16*b^3 + 64*a^17*b^2))/(a^2*f) - (atan((((tan(e + f*
x)*(128*a^3*b^18 + 1984*a^4*b^17 + 13840*a^5*b^16 + 57680*a^6*b^15 + 161280*a^7*b^14 + 322560*a^8*b^13 + 48092
8*a^9*b^12 + 550560*a^10*b^11 + 494400*a^11*b^10 + 352640*a^12*b^9 + 199696*a^13*b^8 + 88144*a^14*b^7 + 29120*
a^15*b^6 + 6720*a^16*b^5 + 960*a^17*b^4 + 64*a^18*b^3) - ((-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(64*a^6*b^17 + 15
36*a^7*b^16 + 13952*a^8*b^15 + 71040*a^9*b^14 + 235968*a^10*b^13 + 551936*a^11*b^12 + 948992*a^12*b^11 + 12291
84*a^13*b^10 + 1214400*a^14*b^9 + 918016*a^15*b^8 + 528000*a^16*b^7 + 227456*a^17*b^6 + 71232*a^18*b^5 + 15360
*a^19*b^4 + 2048*a^20*b^3 + 128*a^21*b^2 - (tan(e + f*x)*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(512*a^7*b^18 + 79
36*a^8*b^17 + 57600*a^9*b^16 + 259840*a^10*b^15 + 815360*a^11*b^14 + 1886976*a^12*b^13 + 3331328*a^13*b^12 + 4
576000*a^14*b^11 + 4942080*a^15*b^10 + 4209920*a^16*b^9 + 2818816*a^17*b^8 + 1467648*a^18*b^7 + 582400*a^19*b^
6 + 170240*a^20*b^5 + 34560*a^21*b^4 + 4352*a^22*b^3 + 256*a^23*b^2))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6
+ 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2))))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 +
35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2)))*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*1i)/(4*(7*a^8*b + a^9 + a^2*b^7 +
7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2)) + ((tan(e + f*x)*(128*a^3*b^18 + 1984*a^4*b^17
+ 13840*a^5*b^16 + 57680*a^6*b^15 + 161280*a^7*b^14 + 322560*a^8*b^13 + 480928*a^9*b^12 + 550560*a^10*b^11 +
494400*a^11*b^10 + 352640*a^12*b^9 + 199696*a^13*b^8 + 88144*a^14*b^7 + 29120*a^15*b^6 + 6720*a^16*b^5 + 960*a
^17*b^4 + 64*a^18*b^3) + ((-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(64*a^6*b^17 + 1536*a^7*b^16 + 13952*a^8*b^15 + 7
1040*a^9*b^14 + 235968*a^10*b^13 + 551936*a^11*b^12 + 948992*a^12*b^11 + 1229184*a^13*b^10 + 1214400*a^14*b^9
+ 918016*a^15*b^8 + 528000*a^16*b^7 + 227456*a^17*b^6 + 71232*a^18*b^5 + 15360*a^19*b^4 + 2048*a^20*b^3 + 128*
a^21*b^2 + (tan(e + f*x)*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(512*a^7*b^18 + 7936*a^8*b^17 + 57600*a^9*b^16 + 2
59840*a^10*b^15 + 815360*a^11*b^14 + 1886976*a^12*b^13 + 3331328*a^13*b^12 + 4576000*a^14*b^11 + 4942080*a^15*
b^10 + 4209920*a^16*b^9 + 2818816*a^17*b^8 + 1467648*a^18*b^7 + 582400*a^19*b^6 + 170240*a^20*b^5 + 34560*a^21
*b^4 + 4352*a^22*b^3 + 256*a^23*b^2))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a
^6*b^3 + 21*a^7*b^2))))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^
7*b^2)))*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*1i)/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*
b^4 + 35*a^6*b^3 + 21*a^7*b^2)))/(304*a^4*b^15 - 208*a^3*b^16 - 32*a^2*b^17 + 7040*a^5*b^14 + 31200*a^6*b^13 +
75936*a^7*b^12 + 118944*a^8*b^11 + 126528*a^9*b^10 + 92640*a^10*b^9 + 46000*a^11*b^8 + 14768*a^12*b^7 + 2752*
a^13*b^6 + 224*a^14*b^5 + ((tan(e + f*x)*(128*a^3*b^18 + 1984*a^4*b^17 + 13840*a^5*b^16 + 57680*a^6*b^15 + 161
280*a^7*b^14 + 322560*a^8*b^13 + 480928*a^9*b^12 + 550560*a^10*b^11 + 494400*a^11*b^10 + 352640*a^12*b^9 + 199
696*a^13*b^8 + 88144*a^14*b^7 + 29120*a^15*b^6 + 6720*a^16*b^5 + 960*a^17*b^4 + 64*a^18*b^3) - ((-b^5*(a + b)^
7)^(1/2)*(7*a + 2*b)*(64*a^6*b^17 + 1536*a^7*b^16 + 13952*a^8*b^15 + 71040*a^9*b^14 + 235968*a^10*b^13 + 55193
6*a^11*b^12 + 948992*a^12*b^11 + 1229184*a^13*b^10 + 1214400*a^14*b^9 + 918016*a^15*b^8 + 528000*a^16*b^7 + 22
7456*a^17*b^6 + 71232*a^18*b^5 + 15360*a^19*b^4 + 2048*a^20*b^3 + 128*a^21*b^2 - (tan(e + f*x)*(-b^5*(a + b)^7
)^(1/2)*(7*a + 2*b)*(512*a^7*b^18 + 7936*a^8*b^17 + 57600*a^9*b^16 + 259840*a^10*b^15 + 815360*a^11*b^14 + 188
6976*a^12*b^13 + 3331328*a^13*b^12 + 4576000*a^14*b^11 + 4942080*a^15*b^10 + 4209920*a^16*b^9 + 2818816*a^17*b
^8 + 1467648*a^18*b^7 + 582400*a^19*b^6 + 170240*a^20*b^5 + 34560*a^21*b^4 + 4352*a^22*b^3 + 256*a^23*b^2))/(4
*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2))))/(4*(7*a^8*b + a^
9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2)))*(-b^5*(a + b)^7)^(1/2)*(7*a + 2
*b))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2)) - ((tan(e +
f*x)*(128*a^3*b^18 + 1984*a^4*b^17 + 13840*a^5*b^16 + 57680*a^6*b^15 + 161280*a^7*b^14 + 322560*a^8*b^13 + 48
0928*a^9*b^12 + 550560*a^10*b^11 + 494400*a^11*b^10 + 352640*a^12*b^9 + 199696*a^13*b^8 + 88144*a^14*b^7 + 291
20*a^15*b^6 + 6720*a^16*b^5 + 960*a^17*b^4 + 64*a^18*b^3) + ((-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(64*a^6*b^17 +
1536*a^7*b^16 + 13952*a^8*b^15 + 71040*a^9*b^14 + 235968*a^10*b^13 + 551936*a^11*b^12 + 948992*a^12*b^11 + 12
29184*a^13*b^10 + 1214400*a^14*b^9 + 918016*a^15*b^8 + 528000*a^16*b^7 + 227456*a^17*b^6 + 71232*a^18*b^5 + 15
360*a^19*b^4 + 2048*a^20*b^3 + 128*a^21*b^2 + (tan(e + f*x)*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*(512*a^7*b^18 +
7936*a^8*b^17 + 57600*a^9*b^16 + 259840*a^10*b^15 + 815360*a^11*b^14 + 1886976*a^12*b^13 + 3331328*a^13*b^12
+ 4576000*a^14*b^11 + 4942080*a^15*b^10 + 4209920*a^16*b^9 + 2818816*a^17*b^8 + 1467648*a^18*b^7 + 582400*a^19
*b^6 + 170240*a^20*b^5 + 34560*a^21*b^4 + 4352*a^22*b^3 + 256*a^23*b^2))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b
^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2))))/(4*(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^
5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2)))*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b))/(4*(7*a^8*b + a^9 + a^2*b^7 +
7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2))))*(-b^5*(a + b)^7)^(1/2)*(7*a + 2*b)*1i)/(2*f*
(7*a^8*b + a^9 + a^2*b^7 + 7*a^3*b^6 + 21*a^4*b^5 + 35*a^5*b^4 + 35*a^6*b^3 + 21*a^7*b^2))
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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(cot(f*x+e)**4/(a+b*sec(f*x+e)**2)**2,x)
[Out]
Timed out
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