\(\int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) [838]

Optimal result

Integrand size = 23, antiderivative size = 385 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]

[Out]

1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a-b)*(a^2+4*a*b+b^2)*
arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+cot(d*x+c)-2^(1/2)*cot
(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+
b^2)^3/d*2^(1/2)-1/4*(a^4+18*a^2*b^2-15*b^4)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))*a^(1/2)/b^(3/2)/(a^2+b^2
)^3/d+1/2*a*cot(d*x+c)^(1/2)/(a^2+b^2)/d/(b+a*cot(d*x+c))^2-1/4*a*(a^2-7*b^2)*cot(d*x+c)^(1/2)/b/(a^2+b^2)^2/d
/(b+a*cot(d*x+c))

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3754, 3649, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac {a \sqrt {\cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d \left (a^2+b^2\right )^3} \]

[In]

Int[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

-(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(
a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[a]*(a^4 + 18*a^2*
b^2 - 15*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*(a^2 + b^2)^3*d) + (a*Sqrt[Cot[c + d*x]
])/(2*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (a*(a^2 - 7*b^2)*Sqrt[Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(b + a
*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*
(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]
*(a^2 + b^2)^3*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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

Rule 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rule 3754

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {\cot (c+d x)}}{(b+a \cot (c+d x))^3} \, dx \\ & = \frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {\int \frac {-\frac {a}{2}-2 b \cot (c+d x)+\frac {3}{2} a \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )} \\ & = \frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} a \left (a^2+9 b^2\right )-2 b \left (a^2-b^2\right ) \cot (c+d x)+\frac {1}{4} a \left (a^2-7 b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b \left (a^2+b^2\right )^2} \\ & = \frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\int \frac {-2 a b \left (a^2-3 b^2\right )-2 b^2 \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 b \left (a^2+b^2\right )^3}+\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{8 b \left (a^2+b^2\right )^3} \\ & = \frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {2 a b \left (a^2-3 b^2\right )+2 b^2 \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b \left (a^2+b^2\right )^3 d}+\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{8 b \left (a^2+b^2\right )^3 d} \\ & = \frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 b \left (a^2+b^2\right )^3 d} \\ & = -\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d} \\ & = -\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a \sqrt {\cot (c+d x)}}{2 \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2-7 b^2\right ) \sqrt {\cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 2.72 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {24 \sqrt {a} \sqrt {b} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )-24 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}+\frac {24 \sqrt {a} \sqrt {b} \left (a^2+b^2\right ) \left (-\sqrt {a} \sqrt {b} \sqrt {\cot (c+d x)}+\arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right ) (b+a \cot (c+d x))\right )}{b+a \cot (c+d x)}+8 b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )+\frac {8 a^2 \left (a^2+b^2\right )^2 \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},-\frac {a \cot (c+d x)}{b}\right )}{b^3}+3 a \left (a^2-3 b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{12 \left (a^2+b^2\right )^3 d} \]

[In]

Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

-1/12*(24*Sqrt[a]*Sqrt[b]*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] - 24*a*(a^2 - 3*b^2)*Sqrt
[Cot[c + d*x]] + (24*Sqrt[a]*Sqrt[b]*(a^2 + b^2)*(-(Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]]) + ArcTan[(Sqrt[a]*Sqrt
[Cot[c + d*x]])/Sqrt[b]]*(b + a*Cot[c + d*x])))/(b + a*Cot[c + d*x]) + 8*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*H
ypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + (8*a^2*(a^2 + b^2)^2*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/2
, 3, 5/2, -((a*Cot[c + d*x])/b)])/b^3 + 3*a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] -
2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c +
d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/((a^2 + b^2)^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2061\) vs. \(2(337)=674\).

Time = 2.18 (sec) , antiderivative size = 2062, normalized size of antiderivative = 5.36

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2062\)
default	\(\text {Expression too large to display}\)	\(2062\)

[In]

int(1/cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/4/d*(-3*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x
+c)))*a^3*b^3-2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*
x+c)-1))*b^6*tan(d*x+c)^2-2*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^6*tan(d*x+c)^2-2*2^(1/2)*
(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^6*tan(d*x+c)^2+3*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+
c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^4*b^2-2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d
*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^2*b^4+2*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2
)*tan(d*x+c)^(1/2))*a^5*b+6*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^4*b^2-6*2^(1/2)*(a*b)^(1/
2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3-18*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^5*b^2+15*arctan(b*ta
n(d*x+c)^(1/2)/(a*b)^(1/2))*a^3*b^4+(a*b)^(1/2)*tan(d*x+c)^(1/2)*a^6-2*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*ta
n(d*x+c)^(1/2))*a^2*b^4+2*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^5*b+6*2^(1/2)*(a*b)^(1/2)*
arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^4*b^2-6*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3-
2*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^4+2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^
(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^5*b-36*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a
^4*b^3*tan(d*x+c)+30*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^2*b^5*tan(d*x+c)-6*(a*b)^(1/2)*tan(d*x+c)^(1/2)*
a^4*b^2-7*(a*b)^(1/2)*tan(d*x+c)^(1/2)*a^2*b^4-(a*b)^(1/2)*tan(d*x+c)^(3/2)*a^5*b-10*(a*b)^(1/2)*tan(d*x+c)^(3
/2)*a^3*b^3-9*(a*b)^(1/2)*tan(d*x+c)^(3/2)*a*b^5-arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^5*b^2*tan(d*x+c)^2-1
8*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^3*b^4*tan(d*x+c)^2+15*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a*b^6*
tan(d*x+c)^2-2*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^6*b*tan(d*x+c)-arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*
a^7+12*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3*tan(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan
(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^4*tan(d*x+c)-6*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^
5*tan(d*x+c)^2+2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan
(d*x+c)))*a^3*b^3*tan(d*x+c)^2-3*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*ta
n(d*x+c)^(1/2)+tan(d*x+c)))*a*b^5*tan(d*x+c)^2+6*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c
))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^3*b^3*tan(d*x+c)-2*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)
^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a*b^5*tan(d*x+c)+4*2^(1/2)*(a*b)^(1/2)*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))*a^4*b^2*tan(d*x+c)-4*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^5*tan(
d*x+c)+4*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^4*b^2*tan(d*x+c)+12*2^(1/2)*(a*b)^(1/2)*arc
tan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3*tan(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))
*a^2*b^4*tan(d*x+c)-4*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^5*tan(d*x+c)+2*2^(1/2)*(a*b)
^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^4*b^2*tan(d*x+c)
-6*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^
2*b^4*tan(d*x+c)+2*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3*tan(d*x+c)^2+6*2^(1/2)*(a*b)
^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^4*tan(d*x+c)^2-6*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+
c)^(1/2))*a*b^5*tan(d*x+c)^2+2*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b^3*tan(d*x+c)^2+6*
2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^4*tan(d*x+c)^2+3*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1
/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^2*b^4*tan(d*x+c)^2)/(1/tan(d*x+c))
^(5/2)/tan(d*x+c)^(5/2)/(a^2+b^2)^3/b/(a+b*tan(d*x+c))^2/(a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4114 vs. \(2 (337) = 674\).

Time = 0.86 (sec) , antiderivative size = 8255, normalized size of antiderivative = 21.44 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/cot(d*x+c)**(5/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (a^{5} + 18 \, a^{3} b^{2} - 15 \, a b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {a^{3} b + 9 \, a b^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {a^{4} - 7 \, a^{2} b^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + \frac {2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )}}{\tan \left (d x + c\right )} + \frac {a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*((a^5 + 18*a^3*b^2 - 15*a*b^4)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 +
b^7)*sqrt(a*b)) - (2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c
)))) + 2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(
2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^3 - 3*a^2
*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)
- ((a^3*b + 9*a*b^3)/sqrt(tan(d*x + c)) - (a^4 - 7*a^2*b^2)/tan(d*x + c)^(3/2))/(a^4*b^3 + 2*a^2*b^5 + b^7 +
2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)/tan(d*x + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)/tan(d*x + c)^2))/d

Giac [F]

\[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^3*cot(d*x + c)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int(1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3),x)

[Out]

int(1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3), x)