3.311 \(\int \frac {\cot ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)
Optimal. Leaf size=360 \[ -\frac {2 b^7 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {b^6 \sin (c+d x)}{a d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {4 b^5 \left (3 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {x}{a^2}+\frac {(3 a+5 b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac {(3 a-5 b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2} \]
[Out]
x/a^2-2*b^7*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^2/(a-b)^(7/2)/(a+b)^(7/2)/d-4*b^5*(3*a^2-b^2
)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^2/(a-b)^(7/2)/(a+b)^(7/2)/d-1/12*sin(d*x+c)/(a+b)^2/d/
(1-cos(d*x+c))^2-1/12*sin(d*x+c)/(a+b)^2/d/(1-cos(d*x+c))+1/4*(3*a+5*b)*sin(d*x+c)/(a+b)^3/d/(1-cos(d*x+c))+1/
12*sin(d*x+c)/(a-b)^2/d/(1+cos(d*x+c))^2-1/4*(3*a-5*b)*sin(d*x+c)/(a-b)^3/d/(1+cos(d*x+c))+1/12*sin(d*x+c)/(a-
b)^2/d/(1+cos(d*x+c))+b^6*sin(d*x+c)/a/(a^2-b^2)^3/d/(b+a*cos(d*x+c))
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Rubi [A] time = 0.57, antiderivative size = 360, normalized size of antiderivative = 1.00,
number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used
= {3898, 2897, 2650, 2648, 2664, 12, 2659, 208} \[ \frac {b^6 \sin (c+d x)}{a d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {2 b^7 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {4 b^5 \left (3 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {x}{a^2}+\frac {(3 a+5 b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac {(3 a-5 b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]
[Out]
x/a^2 - (2*b^7*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b
^5*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - Si
n[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])^2) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])) + ((3*a +
5*b)*Sin[c + d*x])/(4*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])^2) - (
(3*a - 5*b)*Sin[c + d*x])/(4*(a - b)^3*d*(1 + Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])
) + (b^6*Sin[c + d*x])/(a*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 2648
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 2650
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]
Rule 2664
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]
Rule 2897
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))
Rule 3898
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[(Cos[c + d*x]^
m*(b + a*Sin[c + d*x])^n)/Sin[c + d*x]^(m + n), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[
n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(b+a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac {1}{a^2}+\frac {1}{4 (a-b)^2 (-1-\cos (c+d x))^2}+\frac {3 a-5 b}{4 (a-b)^3 (-1-\cos (c+d x))}+\frac {1}{4 (a+b)^2 (1-\cos (c+d x))^2}+\frac {-3 a-5 b}{4 (a+b)^3 (1-\cos (c+d x))}+\frac {b^6}{a^2 \left (a^2-b^2\right )^2 (-b-a \cos (c+d x))^2}+\frac {2 b^5 \left (3 a^2-b^2\right )}{a^2 \left (a^2-b^2\right )^3 (-b-a \cos (c+d x))}\right ) \, dx\\ &=\frac {x}{a^2}+\frac {(3 a-5 b) \int \frac {1}{-1-\cos (c+d x)} \, dx}{4 (a-b)^3}+\frac {\int \frac {1}{(-1-\cos (c+d x))^2} \, dx}{4 (a-b)^2}+\frac {\int \frac {1}{(1-\cos (c+d x))^2} \, dx}{4 (a+b)^2}-\frac {(3 a+5 b) \int \frac {1}{1-\cos (c+d x)} \, dx}{4 (a+b)^3}+\frac {b^6 \int \frac {1}{(-b-a \cos (c+d x))^2} \, dx}{a^2 \left (a^2-b^2\right )^2}+\frac {\left (2 b^5 \left (3 a^2-b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^3}\\ &=\frac {x}{a^2}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}+\frac {(3 a+5 b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}-\frac {(3 a-5 b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {b^6 \sin (c+d x)}{a \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\int \frac {1}{-1-\cos (c+d x)} \, dx}{12 (a-b)^2}+\frac {\int \frac {1}{1-\cos (c+d x)} \, dx}{12 (a+b)^2}+\frac {b^6 \int \frac {b}{-b-a \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^3}+\frac {\left (4 b^5 \left (3 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right )^3 d}\\ &=\frac {x}{a^2}-\frac {4 b^5 \left (3 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {(3 a+5 b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}-\frac {(3 a-5 b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {b^6 \sin (c+d x)}{a \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^7 \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^3}\\ &=\frac {x}{a^2}-\frac {4 b^5 \left (3 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {(3 a+5 b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}-\frac {(3 a-5 b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {b^6 \sin (c+d x)}{a \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (2 b^7\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right )^3 d}\\ &=\frac {x}{a^2}-\frac {2 b^7 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {4 b^5 \left (3 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {(3 a+5 b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}-\frac {(3 a-5 b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {b^6 \sin (c+d x)}{a \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 2.44, size = 303, normalized size = 0.84 \[ \frac {\sec ^2(c+d x) (a \cos (c+d x)+b) \left (-\frac {48 b^5 \left (b^2-6 a^2\right ) (a \cos (c+d x)+b) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{7/2}}+\frac {24 (c+d x) (a \cos (c+d x)+b)}{a^2}+\frac {24 b^6 \sin (c+d x)}{a (a-b)^3 (a+b)^3}+\frac {4 (7 b-4 a) \tan \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^3}+\frac {4 (4 a+7 b) \cot \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^3}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}\right )}{24 d (a+b \sec (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]
[Out]
((b + a*Cos[c + d*x])*Sec[c + d*x]^2*((24*(c + d*x)*(b + a*Cos[c + d*x]))/a^2 - (48*b^5*(-6*a^2 + b^2)*ArcTanh
[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2*(a^2 - b^2)^(7/2)) + (4*(4*a + 7*b)*(
b + a*Cos[c + d*x])*Cot[(c + d*x)/2])/(a + b)^3 - ((b + a*Cos[c + d*x])*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(
a + b)^2 + (24*b^6*Sin[c + d*x])/(a*(a - b)^3*(a + b)^3) + (4*(-4*a + 7*b)*(b + a*Cos[c + d*x])*Tan[(c + d*x)/
2])/(a - b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a - b)^2))/(24*d*(a + b*Sec[c + d*
x])^2)
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fricas [B] time = 0.66, size = 1481, normalized size = 4.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4/(a+b*sec(d*x+c))^2,x, algorithm="fricas")
[Out]
[1/6*(8*a^7*b^2 - 40*a^5*b^4 + 26*a^3*b^6 + 6*a*b^8 + 2*(4*a^9 - 13*a^7*b^2 + 2*a^5*b^4 + 4*a^3*b^6 + 3*a*b^8)
*cos(d*x + c)^4 - 2*(2*a^8*b - 11*a^6*b^3 + 16*a^4*b^5 - 7*a^2*b^7)*cos(d*x + c)^3 - 3*(6*a^2*b^6 - b^8 - (6*a
^3*b^5 - a*b^7)*cos(d*x + c)^3 - (6*a^2*b^6 - b^8)*cos(d*x + c)^2 + (6*a^3*b^5 - a*b^7)*cos(d*x + c))*sqrt(a^2
- b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*
x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 6*(a^9 - 2*a^7*b^2 - 7*a
^5*b^4 + 6*a^3*b^6 + 2*a*b^8)*cos(d*x + c)^2 + 2*(a^8*b - 8*a^6*b^3 + 13*a^4*b^5 - 6*a^2*b^7)*cos(d*x + c) + 6
*((a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c)^3 + (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^
2*b^7 + b^9)*d*x*cos(d*x + c)^2 - (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c) - (a^8*b
- 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x)*sin(d*x + c))/(((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^
3*b^8)*d*cos(d*x + c)^3 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^2 - (a^11 - 4*
a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) - (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*
b^9)*d)*sin(d*x + c)), 1/3*(4*a^7*b^2 - 20*a^5*b^4 + 13*a^3*b^6 + 3*a*b^8 + (4*a^9 - 13*a^7*b^2 + 2*a^5*b^4 +
4*a^3*b^6 + 3*a*b^8)*cos(d*x + c)^4 - (2*a^8*b - 11*a^6*b^3 + 16*a^4*b^5 - 7*a^2*b^7)*cos(d*x + c)^3 + 3*(6*a^
2*b^6 - b^8 - (6*a^3*b^5 - a*b^7)*cos(d*x + c)^3 - (6*a^2*b^6 - b^8)*cos(d*x + c)^2 + (6*a^3*b^5 - a*b^7)*cos(
d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d*x +
c) - 3*(a^9 - 2*a^7*b^2 - 7*a^5*b^4 + 6*a^3*b^6 + 2*a*b^8)*cos(d*x + c)^2 + (a^8*b - 8*a^6*b^3 + 13*a^4*b^5 -
6*a^2*b^7)*cos(d*x + c) + 3*((a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c)^3 + (a^8*b -
4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x*cos(d*x + c)^2 - (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8
)*d*x*cos(d*x + c) - (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x)*sin(d*x + c))/(((a^11 - 4*a^9*b^2
+ 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^3 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d
*cos(d*x + c)^2 - (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) - (a^10*b - 4*a^8*b^3 +
6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d)*sin(d*x + c))]
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giac [A] time = 0.36, size = 487, normalized size = 1.35 \[ -\frac {\frac {48 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {48 \, {\left (6 \, a^{2} b^{5} - b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 27 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{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}} - \frac {24 \, {\left (d x + c\right )}}{a^{2}} - \frac {15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 27 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4/(a+b*sec(d*x+c))^2,x, algorithm="giac")
[Out]
-1/24*(48*b^6*tan(1/2*d*x + 1/2*c)/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/
2*d*x + 1/2*c)^2 - a - b)) - 48*(6*a^2*b^5 - b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a
*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*sq
rt(-a^2 + b^2)) - (a^4*tan(1/2*d*x + 1/2*c)^3 - 4*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 6*a^2*b^2*tan(1/2*d*x + 1/2*c
)^3 - 4*a*b^3*tan(1/2*d*x + 1/2*c)^3 + b^4*tan(1/2*d*x + 1/2*c)^3 - 15*a^4*tan(1/2*d*x + 1/2*c) + 72*a^3*b*tan
(1/2*d*x + 1/2*c) - 126*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*a*b^3*tan(1/2*d*x + 1/2*c) - 27*b^4*tan(1/2*d*x + 1/
2*c))/(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) - 24*(d*x + c)/a^2 - (15*a*tan(1/
2*d*x + 1/2*c)^2 + 27*b*tan(1/2*d*x + 1/2*c)^2 - a - b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*tan(1/2*d*x + 1/2*c)^
3))/d
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maple [A] time = 0.69, size = 416, normalized size = 1.16 \[ \frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{24 d \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 d \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 b^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3} a \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}-\frac {12 b^{5} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 b^{7} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3} a^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{24 d \left (a +b \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {5 a}{8 d \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {9 b}{8 d \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^4/(a+b*sec(d*x+c))^2,x)
[Out]
1/24/d/(a-b)/(a^2-2*a*b+b^2)*a*tan(1/2*d*x+1/2*c)^3-1/24/d/(a-b)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b-5/8/d/
(a-b)/(a^2-2*a*b+b^2)*a*tan(1/2*d*x+1/2*c)+9/8/d/(a-b)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)*b-2/d*b^6/(a+b)^3/(a
-b)^3/a*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)-12/d*b^5/(a+b)^3/(a-b)^3/((a-b)
*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+2/d*b^7/(a+b)^3/(a-b)^3/a^2/((a-b)*(a+b))^
(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-1/24/d/(a+b)^2/tan(1/2*d*x+1/2*c)^3+5/8/d/(a+b)^3/
tan(1/2*d*x+1/2*c)*a+9/8/d/(a+b)^3/tan(1/2*d*x+1/2*c)*b+2/d/a^2*arctan(tan(1/2*d*x+1/2*c))
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4/(a+b*sec(d*x+c))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?
________________________________________________________________________________________
mupad [B] time = 6.97, size = 8348, normalized size = 23.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(c + d*x)^4/(a + b/cos(c + d*x))^2,x)
[Out]
tan(c/2 + (d*x)/2)^3/(24*d*(a - b)^2) + ((3*a*b^2 - 3*a^2*b + a^3 - b^3)/(3*(a + b)) + (2*tan(c/2 + (d*x)/2)^2
*(11*a^3*b - 31*a*b^3 - 8*a^4 + 13*b^4 + 15*a^2*b^2))/(3*(a + b)^2) - (tan(c/2 + (d*x)/2)^4*(11*a^5*b - 9*a*b^
5 - 5*a^6 + 16*b^6 + 31*a^2*b^4 - 34*a^3*b^3 + 6*a^4*b^2))/(a*(a + b)^3))/(d*(tan(c/2 + (d*x)/2)^5*(8*a^4 - 32
*a^3*b - 32*a*b^3 + 8*b^4 + 48*a^2*b^2) - tan(c/2 + (d*x)/2)^3*(16*a*b^3 - 16*a^3*b + 8*a^4 - 8*b^4))) + (tan(
c/2 + (d*x)/2)*((16*a*b + 8*a^2 - 24*b^2)/(64*(a - b)^4) - 3/(4*(a - b)^2)))/d + (2*atan(((tan(c/2 + (d*x)/2)*
(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*
b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a^11*b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*
b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000*a^17*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^2
0*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 373632*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*
a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 4480*a^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 -
384*a^34*b^2) - ((32*a^38 - 32*a^37*b - 32*a^6*b^32 - 32*a^7*b^31 + 640*a^8*b^30 - 4992*a^10*b^28 + 2624*a^11
*b^27 + 21504*a^12*b^26 - 19872*a^13*b^25 - 57920*a^14*b^24 + 77472*a^15*b^23 + 100992*a^16*b^22 - 195008*a^17
*b^21 - 107008*a^18*b^20 + 344960*a^19*b^19 + 39424*a^20*b^18 - 446688*a^21*b^17 + 76032*a^22*b^16 + 431904*a^
23*b^15 - 161920*a^24*b^14 - 313984*a^25*b^13 + 167552*a^26*b^12 + 171072*a^27*b^11 - 113664*a^28*b^10 - 68960
*a^29*b^9 + 53568*a^30*b^8 + 20064*a^31*b^7 - 17536*a^32*b^6 - 4032*a^33*b^5 + 3840*a^34*b^4 + 512*a^35*b^3 -
512*a^36*b^2 + (tan(c/2 + (d*x)/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5
760*a^11*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 - 58240*a^15*b^25 + 174720*a^16*b^24 + 104
832*a^17*b^23 - 384384*a^18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^20 + 91520*a^21*b^19 - 823680*a^22*b^18 +
823680*a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b^13 + 384384*a^28*b^12 - 104832*a^29*b^11
- 174720*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920
*a^36*b^4 - 896*a^37*b^3 - 128*a^38*b^2)*1i)/a^2)*1i)/a^2)/a^2 + (tan(c/2 + (d*x)/2)*(32*a^36 - 96*a^35*b + 64
*a^3*b^33 - 128*a^4*b^32 - 1056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^9*b^27 + 6
7200*a^10*b^26 + 77760*a^11*b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^15*b^21 -
580608*a^16*b^20 + 96000*a^17*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640*a^21*b^15
+ 319520*a^22*b^14 - 373632*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116480*a^27*b
^9 + 40320*a^29*b^7 - 4480*a^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2) - ((32*a^3
7*b - 32*a^38 + 32*a^6*b^32 + 32*a^7*b^31 - 640*a^8*b^30 + 4992*a^10*b^28 - 2624*a^11*b^27 - 21504*a^12*b^26 +
19872*a^13*b^25 + 57920*a^14*b^24 - 77472*a^15*b^23 - 100992*a^16*b^22 + 195008*a^17*b^21 + 107008*a^18*b^20
- 344960*a^19*b^19 - 39424*a^20*b^18 + 446688*a^21*b^17 - 76032*a^22*b^16 - 431904*a^23*b^15 + 161920*a^24*b^1
4 + 313984*a^25*b^13 - 167552*a^26*b^12 - 171072*a^27*b^11 + 113664*a^28*b^10 + 68960*a^29*b^9 - 53568*a^30*b^
8 - 20064*a^31*b^7 + 17536*a^32*b^6 + 4032*a^33*b^5 - 3840*a^34*b^4 - 512*a^35*b^3 + 512*a^36*b^2 + (tan(c/2 +
(d*x)/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^1
2*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 - 58240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^
18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^20 + 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*
a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b^13 + 384384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 582
40*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3
- 128*a^38*b^2)*1i)/a^2)*1i)/a^2)/a^2)/(((tan(c/2 + (d*x)/2)*(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32
- 1056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a
^11*b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000
*a^17*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 37
3632*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 44
80*a^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2) - ((32*a^38 - 32*a^37*b - 32*a^6*b
^32 - 32*a^7*b^31 + 640*a^8*b^30 - 4992*a^10*b^28 + 2624*a^11*b^27 + 21504*a^12*b^26 - 19872*a^13*b^25 - 57920
*a^14*b^24 + 77472*a^15*b^23 + 100992*a^16*b^22 - 195008*a^17*b^21 - 107008*a^18*b^20 + 344960*a^19*b^19 + 394
24*a^20*b^18 - 446688*a^21*b^17 + 76032*a^22*b^16 + 431904*a^23*b^15 - 161920*a^24*b^14 - 313984*a^25*b^13 + 1
67552*a^26*b^12 + 171072*a^27*b^11 - 113664*a^28*b^10 - 68960*a^29*b^9 + 53568*a^30*b^8 + 20064*a^31*b^7 - 175
36*a^32*b^6 - 4032*a^33*b^5 + 3840*a^34*b^4 + 512*a^35*b^3 - 512*a^36*b^2 + (tan(c/2 + (d*x)/2)*(64*a^39*b - 6
4*a^7*b^33 + 128*a^8*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27
- 58240*a^14*b^26 - 58240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^18*b^22 - 128128*a^19*b^
21 + 640640*a^20*b^20 + 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*
b^14 + 128128*a^27*b^13 + 384384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32
*b^8 - 22400*a^33*b^7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3 - 128*a^38*b^2)*1i)/a^2)
*1i)/a^2)*1i)/a^2 - ((tan(c/2 + (d*x)/2)*(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1056*a^5*b^31 + 2
080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a^11*b^25 - 194240*a^
12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000*a^17*b^19 + 636160*
a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 373632*a^23*b^13 - 138
880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 4480*a^30*b^6 - 9600*a
^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2) - ((32*a^37*b - 32*a^38 + 32*a^6*b^32 + 32*a^7*b^31 -
640*a^8*b^30 + 4992*a^10*b^28 - 2624*a^11*b^27 - 21504*a^12*b^26 + 19872*a^13*b^25 + 57920*a^14*b^24 - 77472*a
^15*b^23 - 100992*a^16*b^22 + 195008*a^17*b^21 + 107008*a^18*b^20 - 344960*a^19*b^19 - 39424*a^20*b^18 + 44668
8*a^21*b^17 - 76032*a^22*b^16 - 431904*a^23*b^15 + 161920*a^24*b^14 + 313984*a^25*b^13 - 167552*a^26*b^12 - 17
1072*a^27*b^11 + 113664*a^28*b^10 + 68960*a^29*b^9 - 53568*a^30*b^8 - 20064*a^31*b^7 + 17536*a^32*b^6 + 4032*a
^33*b^5 - 3840*a^34*b^4 - 512*a^35*b^3 + 512*a^36*b^2 + (tan(c/2 + (d*x)/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8
*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 -
58240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^2
0 + 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b
^13 + 384384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^
7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3 - 128*a^38*b^2)*1i)/a^2)*1i)/a^2)*1i)/a^2 +
64*a^2*b^32 - 256*a^3*b^31 - 960*a^4*b^30 + 3840*a^5*b^29 + 6144*a^6*b^28 - 23168*a^7*b^27 - 25088*a^8*b^26 +
78784*a^9*b^25 + 76800*a^10*b^24 - 173760*a^11*b^23 - 183168*a^12*b^22 + 269952*a^13*b^21 + 334080*a^14*b^20 -
314880*a^15*b^19 - 453888*a^16*b^18 + 291456*a^17*b^17 + 449856*a^18*b^16 - 221568*a^19*b^15 - 318400*a^20*b^
14 + 136960*a^21*b^13 + 155904*a^22*b^12 - 64896*a^23*b^11 - 49920*a^24*b^10 + 21440*a^25*b^9 + 9344*a^26*b^8
- 4288*a^27*b^7 - 768*a^28*b^6 + 384*a^29*b^5)))/(a^2*d) + (b^5*atan(((b^5*(tan(c/2 + (d*x)/2)*(32*a^36 - 96*a
^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^
9*b^27 + 67200*a^10*b^26 + 77760*a^11*b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^
15*b^21 - 580608*a^16*b^20 + 96000*a^17*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640
*a^21*b^15 + 319520*a^22*b^14 - 373632*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116
480*a^27*b^9 + 40320*a^29*b^7 - 4480*a^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2)
- (b^5*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*(32*a^38 - 32*a^37*b - 32*a^6*b^32 - 32*a^7*b^31 + 640*a^8*b^
30 - 4992*a^10*b^28 + 2624*a^11*b^27 + 21504*a^12*b^26 - 19872*a^13*b^25 - 57920*a^14*b^24 + 77472*a^15*b^23 +
100992*a^16*b^22 - 195008*a^17*b^21 - 107008*a^18*b^20 + 344960*a^19*b^19 + 39424*a^20*b^18 - 446688*a^21*b^1
7 + 76032*a^22*b^16 + 431904*a^23*b^15 - 161920*a^24*b^14 - 313984*a^25*b^13 + 167552*a^26*b^12 + 171072*a^27*
b^11 - 113664*a^28*b^10 - 68960*a^29*b^9 + 53568*a^30*b^8 + 20064*a^31*b^7 - 17536*a^32*b^6 - 4032*a^33*b^5 +
3840*a^34*b^4 + 512*a^35*b^3 - 512*a^36*b^2 + (b^5*tan(c/2 + (d*x)/2)*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2
)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^12*b^28 +
22400*a^13*b^27 - 58240*a^14*b^26 - 58240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^18*b^22
- 128128*a^19*b^21 + 640640*a^20*b^20 + 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*a^25*b^1
5 - 640640*a^26*b^14 + 128128*a^27*b^13 + 384384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 58240*a^31*
b^9 + 58240*a^32*b^8 - 22400*a^33*b^7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3 - 128*a^
38*b^2))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2)))/
(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2))*(6*a^2 - b
^2)*((a + b)^7*(a - b)^7)^(1/2)*1i)/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 2
1*a^12*b^4 - 7*a^14*b^2) + (b^5*(tan(c/2 + (d*x)/2)*(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1056*a
^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a^11*b^25
- 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000*a^17*b^1
9 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 373632*a^23
*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 4480*a^30*b
^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2) - (b^5*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^
(1/2)*(32*a^37*b - 32*a^38 + 32*a^6*b^32 + 32*a^7*b^31 - 640*a^8*b^30 + 4992*a^10*b^28 - 2624*a^11*b^27 - 2150
4*a^12*b^26 + 19872*a^13*b^25 + 57920*a^14*b^24 - 77472*a^15*b^23 - 100992*a^16*b^22 + 195008*a^17*b^21 + 1070
08*a^18*b^20 - 344960*a^19*b^19 - 39424*a^20*b^18 + 446688*a^21*b^17 - 76032*a^22*b^16 - 431904*a^23*b^15 + 16
1920*a^24*b^14 + 313984*a^25*b^13 - 167552*a^26*b^12 - 171072*a^27*b^11 + 113664*a^28*b^10 + 68960*a^29*b^9 -
53568*a^30*b^8 - 20064*a^31*b^7 + 17536*a^32*b^6 + 4032*a^33*b^5 - 3840*a^34*b^4 - 512*a^35*b^3 + 512*a^36*b^2
+ (b^5*tan(c/2 + (d*x)/2)*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b^32 +
896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 - 58240*
a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^20 + 915
20*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b^13 + 3
84384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^7 - 134
40*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3 - 128*a^38*b^2))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21
*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2)))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10
+ 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2))*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*1i)/(a^16 -
a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2))/(64*a^2*b^32 - 25
6*a^3*b^31 - 960*a^4*b^30 + 3840*a^5*b^29 + 6144*a^6*b^28 - 23168*a^7*b^27 - 25088*a^8*b^26 + 78784*a^9*b^25 +
76800*a^10*b^24 - 173760*a^11*b^23 - 183168*a^12*b^22 + 269952*a^13*b^21 + 334080*a^14*b^20 - 314880*a^15*b^1
9 - 453888*a^16*b^18 + 291456*a^17*b^17 + 449856*a^18*b^16 - 221568*a^19*b^15 - 318400*a^20*b^14 + 136960*a^21
*b^13 + 155904*a^22*b^12 - 64896*a^23*b^11 - 49920*a^24*b^10 + 21440*a^25*b^9 + 9344*a^26*b^8 - 4288*a^27*b^7
- 768*a^28*b^6 + 384*a^29*b^5 + (b^5*(tan(c/2 + (d*x)/2)*(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1
056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 - 15360*a^8*b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a^11*
b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 393792*a^14*b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000*a^1
7*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 - 522720*a^20*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 373632
*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^11 + 36096*a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 4480*a
^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408*a^33*b^3 - 384*a^34*b^2) - (b^5*(6*a^2 - b^2)*((a + b)^7*(a - b
)^7)^(1/2)*(32*a^38 - 32*a^37*b - 32*a^6*b^32 - 32*a^7*b^31 + 640*a^8*b^30 - 4992*a^10*b^28 + 2624*a^11*b^27 +
21504*a^12*b^26 - 19872*a^13*b^25 - 57920*a^14*b^24 + 77472*a^15*b^23 + 100992*a^16*b^22 - 195008*a^17*b^21 -
107008*a^18*b^20 + 344960*a^19*b^19 + 39424*a^20*b^18 - 446688*a^21*b^17 + 76032*a^22*b^16 + 431904*a^23*b^15
- 161920*a^24*b^14 - 313984*a^25*b^13 + 167552*a^26*b^12 + 171072*a^27*b^11 - 113664*a^28*b^10 - 68960*a^29*b
^9 + 53568*a^30*b^8 + 20064*a^31*b^7 - 17536*a^32*b^6 - 4032*a^33*b^5 + 3840*a^34*b^4 + 512*a^35*b^3 - 512*a^3
6*b^2 + (b^5*tan(c/2 + (d*x)/2)*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b
^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^11*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 - 5
8240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^17*b^23 - 384384*a^18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^20
+ 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b^1
3 + 384384*a^28*b^12 - 104832*a^29*b^11 - 174720*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^7
- 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b^4 - 896*a^37*b^3 - 128*a^38*b^2))/(a^16 - a^2*b^14 + 7*a^4*b^12
- 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2)))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6
*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2))*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2))/(a^16
- a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2) - (b^5*(tan(c/2
+ (d*x)/2)*(32*a^36 - 96*a^35*b + 64*a^3*b^33 - 128*a^4*b^32 - 1056*a^5*b^31 + 2080*a^6*b^30 + 7680*a^7*b^29 -
15360*a^8*b^28 - 31360*a^9*b^27 + 67200*a^10*b^26 + 77760*a^11*b^25 - 194240*a^12*b^24 - 114240*a^13*b^23 + 3
93792*a^14*b^22 + 68096*a^15*b^21 - 580608*a^16*b^20 + 96000*a^17*b^19 + 636160*a^18*b^18 - 300960*a^19*b^17 -
522720*a^20*b^16 + 412640*a^21*b^15 + 319520*a^22*b^14 - 373632*a^23*b^13 - 138880*a^24*b^12 + 243456*a^25*b^
11 + 36096*a^26*b^10 - 116480*a^27*b^9 + 40320*a^29*b^7 - 4480*a^30*b^6 - 9600*a^31*b^5 + 1920*a^32*b^4 + 1408
*a^33*b^3 - 384*a^34*b^2) - (b^5*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*(32*a^37*b - 32*a^38 + 32*a^6*b^32
+ 32*a^7*b^31 - 640*a^8*b^30 + 4992*a^10*b^28 - 2624*a^11*b^27 - 21504*a^12*b^26 + 19872*a^13*b^25 + 57920*a^1
4*b^24 - 77472*a^15*b^23 - 100992*a^16*b^22 + 195008*a^17*b^21 + 107008*a^18*b^20 - 344960*a^19*b^19 - 39424*a
^20*b^18 + 446688*a^21*b^17 - 76032*a^22*b^16 - 431904*a^23*b^15 + 161920*a^24*b^14 + 313984*a^25*b^13 - 16755
2*a^26*b^12 - 171072*a^27*b^11 + 113664*a^28*b^10 + 68960*a^29*b^9 - 53568*a^30*b^8 - 20064*a^31*b^7 + 17536*a
^32*b^6 + 4032*a^33*b^5 - 3840*a^34*b^4 - 512*a^35*b^3 + 512*a^36*b^2 + (b^5*tan(c/2 + (d*x)/2)*(6*a^2 - b^2)*
((a + b)^7*(a - b)^7)^(1/2)*(64*a^39*b - 64*a^7*b^33 + 128*a^8*b^32 + 896*a^9*b^31 - 1920*a^10*b^30 - 5760*a^1
1*b^29 + 13440*a^12*b^28 + 22400*a^13*b^27 - 58240*a^14*b^26 - 58240*a^15*b^25 + 174720*a^16*b^24 + 104832*a^1
7*b^23 - 384384*a^18*b^22 - 128128*a^19*b^21 + 640640*a^20*b^20 + 91520*a^21*b^19 - 823680*a^22*b^18 + 823680*
a^24*b^16 - 91520*a^25*b^15 - 640640*a^26*b^14 + 128128*a^27*b^13 + 384384*a^28*b^12 - 104832*a^29*b^11 - 1747
20*a^30*b^10 + 58240*a^31*b^9 + 58240*a^32*b^8 - 22400*a^33*b^7 - 13440*a^34*b^6 + 5760*a^35*b^5 + 1920*a^36*b
^4 - 896*a^37*b^3 - 128*a^38*b^2))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21
*a^12*b^4 - 7*a^14*b^2)))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4
- 7*a^14*b^2))*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2))/(a^16 - a^2*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^
8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2)))*(6*a^2 - b^2)*((a + b)^7*(a - b)^7)^(1/2)*2i)/(d*(a^16 - a^2
*b^14 + 7*a^4*b^12 - 21*a^6*b^10 + 35*a^8*b^8 - 35*a^10*b^6 + 21*a^12*b^4 - 7*a^14*b^2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**4/(a+b*sec(d*x+c))**2,x)
[Out]
Integral(cot(c + d*x)**4/(a + b*sec(c + d*x))**2, x)
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