\(\int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [631]

Optimal result

Integrand size = 35, antiderivative size = 300 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {(i a-b)^{5/2} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 b^{5/2} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 a (2 A b+a B) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d} \]

[Out]

-(I*a-b)^(5/2)*(I*A-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+2*b^(5/2)*B*arctanh(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)*(I*A+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)*ta
n(d*x+c)^(1/2)/d-2*a*(2*A*b+B*a)*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1/2)/d-2/3*a*A*cot(d*x+c)^(3/2)*(a+b*tan(d
*x+c))^(3/2)/d

Rubi [A] (verified)

Time = 2.90 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {4326, 3686, 3726, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a (a B+2 A b) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b^{5/2} B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \]

[In]

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

[Out]

-(((I*a - b)^(5/2)*(I*A - 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) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqr
t[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*(I*A + 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*a*(2*A*b + a*B)*Sqrt[Cot[c + d*x]]*Sqr
t[a + b*Tan[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2))/(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 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 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 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 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 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 3736

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] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2)), x], x, Tan[
e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [C] (verified)

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

Time = 5.72 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.39 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\cot ^{\frac {3}{2}}(c+d x) \left (i a b (A+i B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b \tan (c+d x)}{a}\right ) \sqrt {a+b \tan (c+d x)}-a b (i A+B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b \tan (c+d x)}{a}\right ) \sqrt {a+b \tan (c+d x)}+(i a+b) (A-i B) \sqrt {1+\frac {b \tan (c+d x)}{a}} \left (3 \sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {3}{2}}(c+d x)+(i a+(-3 a+4 i b) \tan (c+d x)) \sqrt {a+b \tan (c+d x)}\right )+(i a-b) (A+i B) \sqrt {1+\frac {b \tan (c+d x)}{a}} \left (3 \sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {3}{2}}(c+d x)+(i a+(3 a+4 i b) \tan (c+d x)) \sqrt {a+b \tan (c+d x)}\right )\right )}{3 d \sqrt {1+\frac {b \tan (c+d x)}{a}}} \]

[In]

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

[Out]

(Cot[c + d*x]^(3/2)*(I*a*b*(A + I*B)*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*Tan[c + d*x])/a)]*Sqrt[a + b*Tan
[c + d*x]] - a*b*(I*A + B)*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*Tan[c + d*x])/a)]*Sqrt[a + b*Tan[c + d*x]]
+ (I*a + b)*(A - I*B)*Sqrt[1 + (b*Tan[c + d*x])/a]*(3*(-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a
+ I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + (I*a + (-3*a + (4*I)*b)*Tan[c + d*x]
)*Sqrt[a + b*Tan[c + d*x]]) + (I*a - b)*(A + I*B)*Sqrt[1 + (b*Tan[c + d*x])/a]*(3*(-1)^(1/4)*(a + I*b)^(3/2)*A
rcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + (I*a + (3*a
+ (4*I)*b)*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]])))/(3*d*Sqrt[1 + (b*Tan[c + d*x])/a])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2437\) vs. \(2(248)=496\).

Time = 1.41 (sec) , antiderivative size = 2438, normalized size of antiderivative = 8.13

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2438\)
default	\(\text {Expression too large to display}\)	\(2438\)

[In]

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

[Out]

1/12/d*((b+a*cot(d*x+c))/cot(d*x+c))^(1/2)*cot(d*x+c)^(1/2)*(-36*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+36*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^3*b-12*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*b^3-36*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^3*b+12*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*b^3-12*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^3+12*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^3-24*B*a^3*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(
1/2)-2*b)^(1/2)+36*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^2-8*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)+3*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)*b^3-3*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)*b^3-3*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^3+3*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^3-24*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^2*b+24*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^2*b-56*A*a^2*b*(b+a*cot(d*x+c))^(1/2)*(2
*(a^2+b^2)^(1/2)-2*b)^(1/2)+12*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*b^2-12*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*b^2+9*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*b^2-9*B*ln(a*c
ot(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+3*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^2-3*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)*b^2-9*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-3*A*ln(a*c
ot(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+3*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)*
b^2+9*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-12*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^4+12*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^4-6*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*b+6*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*b+24*B*b^(5/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a*(
2*(a^2+b^2)^(1/2)-2*b)^(1/2))/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. 17767 vs. \(2 (242) = 484\).

Time = 7.26 (sec) , antiderivative size = 35564, normalized size of antiderivative = 118.55 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate(cot(d*x+c)^(5/2)*(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)^(5/2), x)

Giac [F(-1)]

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

[In]

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

[In]

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

[Out]

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