\(\int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) [832]

Optimal result

Integrand size = 23, antiderivative size = 357 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]

[Out]

a^(5/2)*(3*a^2+7*b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/b^(5/2)/(a^2+b^2)^2/d-1/2*(a^2-2*a*b-b^2)*arcta
n(-1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a
^2+b^2)^2/d*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a
^2+2*a*b-b^2)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+(3*a^2+2*b^2)/b^2/(a^2+b^2)/d/co
t(d*x+c)^(1/2)-a^2/b/(a^2+b^2)/d/(b+a*cot(d*x+c))/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 357, 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, 3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}+\frac {3 a^2+2 b^2}{b^2 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2} \]

[In]

Int[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*
ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(5/2)*(3*a^2 + 7*b^2)*ArcTan[(Sqrt[a]*S
qrt[Cot[c + d*x]])/Sqrt[b]])/(b^(5/2)*(a^2 + b^2)^2*d) + (3*a^2 + 2*b^2)/(b^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]
) - a^2/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[
Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c +
d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*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 3650

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

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 {1}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2} \, dx \\ & = -\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\int \frac {\frac {1}{2} \left (-3 a^2-2 b^2\right )+a b \cot (c+d x)-\frac {3}{2} a^2 \cot ^2(c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {\frac {1}{4} a \left (3 a^2+4 b^2\right )+\frac {1}{2} b^3 \cot (c+d x)+\frac {1}{4} a \left (3 a^2+2 b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )} \\ & = \frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {a b^3-\frac {1}{2} b^2 \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )^2}-\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2} \\ & = \frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {4 \text {Subst}\left (\int \frac {-a b^3+\frac {1}{2} b^2 \left (a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 b^2 \left (a^2+b^2\right )^2 d} \\ & = \frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}+\frac {\left (a^2+2 a b-b^2\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 )^2 d}+\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d} \\ & = \frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}-\frac {\left (a^2+2 a b-b^2\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 )^2 d}-\frac {\left (a^2+2 a b-b^2\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 )^2 d} \\ & = \frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 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 )^2 d}+\frac {\left (a^2+2 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 )^2 d}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}+\frac {\left (a^2-2 a b-b^2\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 )^2 d} \\ & = \frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 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 )^2 d}+\frac {\left (a^2+2 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 )^2 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.61 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {8 a^2 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {a \cot (c+d x)}{b}\right )+4 a^2 \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},-\frac {a \cot (c+d x)}{b}\right )+b^2 \left (-4 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} a b \sqrt {\cot (c+d x)} \left (-2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{2 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}} \]

[In]

Integrate[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1053\) vs. \(2(315)=630\).

Time = 1.82 (sec) , antiderivative size = 1054, normalized size of antiderivative = 2.95

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1054\)
default	\(\text {Expression too large to display}\)	\(1054\)

[In]

int(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/4/d*(-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))*2^(1/2)*(a*b)^(1
/2)*a^2*b^3*tan(d*x+c)-2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^2*b^3*tan(d*x+c)+4*arctan(1+
2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4*tan(d*x+c)-2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(
a*b)^(1/2)*a^2*b^3*tan(d*x+c)+4*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4*tan(d*x+c)+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)))*2^(1/2)*(a*b)^(1/2)*a*b^4*ta
n(d*x+c)+28*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^4*b^2-12*(a*b)^(1/2)*tan(d*x+c)^(1/2)*a^5-8*(a*b)^(1/2)*t
an(d*x+c)^(3/2)*b^5-8*(a*b)^(1/2)*tan(d*x+c)^(3/2)*a^4*b-16*(a*b)^(1/2)*tan(d*x+c)^(3/2)*a^2*b^3+12*arctan(b*t
an(d*x+c)^(1/2)/(a*b)^(1/2))*a^5*b*tan(d*x+c)+28*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^3*b^3*tan(d*x+c)-20*
(a*b)^(1/2)*tan(d*x+c)^(1/2)*a^3*b^2-8*(a*b)^(1/2)*tan(d*x+c)^(1/2)*a*b^4+12*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(
1/2))*a^6-2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^3*b^2+4*arctan(1+2^(1/2)*tan(d*x+c)^(1/2)
)*2^(1/2)*(a*b)^(1/2)*a^2*b^3+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4-2*arctan(-1+2^(1/
2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^3*b^2+4*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^2
*b^3+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4+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)))*2^(1/2)*(a*b)^(1/2)*a^2*b^3+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))*2^(1/2)*(a*b)^(1/2)*b^5*tan(d*x+c)+2*arctan(1+2^(1/2)*tan(d*x
+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*b^5*tan(d*x+c)+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*b^5*ta
n(d*x+c)-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))*2^(1/2)*(a*b)^(1
/2)*a^3*b^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))*2^(1/2)*(a*b)
^(1/2)*a*b^4)/(1/tan(d*x+c))^(7/2)/tan(d*x+c)^(7/2)/b^2/(a^2+b^2)^2/(a+b*tan(d*x+c))/(a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2717 vs. \(2 (315) = 630\).

Time = 0.63 (sec) , antiderivative size = 5459, normalized size of antiderivative = 15.29 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(1/cot(d*x+c)**(7/2)/(a+b*tan(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, a^{5} + 7 \, a^{3} b^{2}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, a^{2} b + 2 \, b^{3} + \frac {3 \, a^{3} + 2 \, a b^{2}}{\tan \left (d x + c\right )}\right )}}{\frac {a^{2} b^{3} + b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {a^{3} b^{2} + a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}}{4 \, d} \]

[In]

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^2*cot(d*x + c)^(7/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int(1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2),x)

[Out]

int(1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2), x)