\(\int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [630]

Optimal result

Integrand size = 35, antiderivative size = 287 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {(i a-b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {(i a+b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d} \]

[Out]

-(I*a-b)^(5/2)*(A+I*B)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+
c)^(1/2)/d+(I*a+b)^(5/2)*(A-I*B)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/
2)*tan(d*x+c)^(1/2)/d-2/15*a*(8*A*b+5*B*a)*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/d+2/15*(15*A*a^2-23*A*b^2-3
5*B*a*b)*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1/2)/d-2/5*a*A*cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^(3/2)/d

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4326, 3686, 3726, 3730, 3697, 3696, 95, 209, 212} \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 \left (15 a^2 A-35 a b B-23 A b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {(-b+i a)^{5/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(b+i a)^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a (5 a B+8 A b) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d} \]

[In]

Int[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

-(((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c +
d*x]]*Sqrt[Tan[c + d*x]])/d) + ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a +
b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Sqrt[Cot[c + d
*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (2*a*(8*A*b + 5*a*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*
d) - (2*a*A*Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2))/(5*d)

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 209

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

Rule 212

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

Rule 3686

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

Rule 3696

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]

Rule 3697

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

Rule 3726

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*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

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 4326

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}+\frac {1}{5} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {1}{2} a (8 A b+5 a B)-\frac {5}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (2 a A-5 b B) \tan ^2(c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}+\frac {1}{15} \left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (15 a^2 A-23 A b^2-35 a b B\right )-\frac {15}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac {1}{4} b \left (22 a A b+10 a^2 B-15 b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {15}{8} a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {15}{8} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{15 a} \\ & = \frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac {1}{2} \left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (4 \left (\frac {15}{8} a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {15}{8} i a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{15 a} \\ & = \frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac {\left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left (4 \left (\frac {15}{8} a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {15}{8} i a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{15 a d} \\ & = \frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac {\left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (8 \left (\frac {15}{8} a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {15}{8} i a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{15 a d} \\ & = -\frac {(i a-b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {(i a+b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (15 a^2 A-23 A b^2-35 a b B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a (8 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{15 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.10 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.12 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\cot ^{\frac {5}{2}}(c+d x) \left (60 \sqrt [4]{-1} \left ((-a+i b)^{5/2} (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+(a+i b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right ) \tan ^{\frac {5}{2}}(c+d x)+15 b (-2 A b+a B) \sqrt {a+b \tan (c+d x)}-3 \left (8 a^2 A-10 A b^2-15 a b B\right ) \sqrt {a+b \tan (c+d x)}-4 \left (22 a A b+10 a^2 B-15 b^2 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}+8 \left (15 a^2 A-23 A b^2-35 a b B\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}-60 b B (a+b \tan (c+d x))^{3/2}\right )}{60 d} \]

[In]

Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(Cot[c + d*x]^(5/2)*(60*(-1)^(1/4)*((-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c +
d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x
]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) + 15*b*(-2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]] - 3*(8*a^2*A
- 10*A*b^2 - 15*a*b*B)*Sqrt[a + b*Tan[c + d*x]] - 4*(22*a*A*b + 10*a^2*B - 15*b^2*B)*Tan[c + d*x]*Sqrt[a + b*T
an[c + d*x]] + 8*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]] - 60*b*B*(a + b*Tan[
c + d*x])^(3/2)))/(60*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2517\) vs. \(2(239)=478\).

Time = 1.62 (sec) , antiderivative size = 2518, normalized size of antiderivative = 8.77

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2518\)
default	\(\text {Expression too large to display}\)	\(2518\)

[In]

int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/60/d*((b+a*cot(d*x+c))/cot(d*x+c))^(1/2)*cot(d*x+c)^(1/2)*(-24*A*a^3*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/
2)-2*b)^(1/2)*cot(d*x+c)^2-40*B*a^3*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x+c)+15*A*ln((b
+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/
2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a^3-15*A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/
2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a^3+15*B*ln((b+a*cot(d*x+c))^(
1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^
(1/2)-2*b)^(1/2)*b^3-15*B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/
2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^3-60*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)
-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*a*b^2+60*A*arctan((2*(b+a*cot(d*x+c)
)^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*a*b^2-184*A*b^2*(b+a*cot
(d*x+c))^(1/2)*a*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)-120*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^
(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*a^2*b+120*B*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2
)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*a^2*b-280*B*a^2*b*(b+a*cot(d*x+c))^(1/2)*(2
*(a^2+b^2)^(1/2)-2*b)^(1/2)-180*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)
^(1/2)-2*b)^(1/2))*a^3*b+60*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/
2)-2*b)^(1/2))*a*b^3+180*A*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-
2*b)^(1/2))*a^3*b-60*A*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)
^(1/2))*a*b^3+180*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1
/2))*a^2*b^2-180*B*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/
2))*a^2*b^2+60*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2)
)*(a^2+b^2)^(1/2)*a^3-60*A*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-
2*b)^(1/2))*(a^2+b^2)^(1/2)*a^3+120*A*a^3*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)-60*B*arctan(((2
*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*a^4+60*B*arctan((2*(b+a*c
ot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*a^4-88*A*a^2*b*(b+a*cot(d*x+c))
^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x+c)-45*A*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a
*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a*b^2+45*A*ln(a*cot
(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*
(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a*b^2+15*B*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-
b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*a^2-15*B*ln((b+
a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2
)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^2-45*B*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/
2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a^2*b-15*B*ln(a
*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1
/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*a^2+15*B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^
2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/
2)*b^2+45*B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^
2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a^2*b+30*A*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)
^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(
1/2)*a*b-30*A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+
b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*a*b)/a/(b+a*cot(d*x+c))^(1/2)/(2*(a^2+b^2)
^(1/2)-2*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 18942 vs. \(2 (233) = 466\).

Time = 4.95 (sec) , antiderivative size = 18942, normalized size of antiderivative = 66.00 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**(7/2)*(a+b*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(7/2), x)

Giac [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int(cot(c + d*x)^(7/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2),x)

[Out]

int(cot(c + d*x)^(7/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2), x)