\(\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [632]

Optimal result

Integrand size = 35, antiderivative size = 301 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(i a-b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b^{3/2} (2 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d} \]

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

Rubi [A] (verified)

Time = 2.80 (sec) , antiderivative size = 301, 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, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(-b+i a)^{5/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {b^{3/2} (5 a B+2 A b) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d} \]

[In]

Int[Cot[c + d*x]^(3/2)*(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 + (b^(3/2)*(2*A*b + 5*a*B)*ArcTanh[(Sqrt[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 + (b*(2*a*A + b*B)*Sqrt[a + b*Tan[
c + d*x]])/(d*Sqrt[Cot[c + d*x]]) - (2*a*A*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/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 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 3728

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

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 {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\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 (4 A b+a B)-\frac {1}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {1}{2} b (2 a A+b B) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} a \left (6 a A b+2 a^2 B-b^2 B\right )-\frac {1}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)+\frac {1}{4} b^2 (2 A b+5 a B) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} a \left (6 a A b+2 a^2 B-b^2 B\right )+\frac {1}{2} \left (-a^3 A+3 a A b^2+3 a^2 b B-b^3 B\right ) x+\frac {1}{4} b^2 (2 A b+5 a B) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {b^2 (2 A b+5 a B)}{4 \sqrt {x} \sqrt {a+b x}}+\frac {3 a^2 A b-A b^3+a^3 B-3 a b^2 B-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x}{2 \sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {3 a^2 A b-A b^3+a^3 B-3 a b^2 B-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 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^2 (2 A b+5 a 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 )}{2 d} \\ & = \frac {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {a^3 A-3 a A b^2-3 a^2 b B+b^3 B+i \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {-a^3 A+3 a A b^2+3 a^2 b B-b^3 B+i \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (b^2 (2 A b+5 a 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 {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}-\frac {\left ((a-i b)^3 (A-i 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 ((a+i b)^3 (A+i B) \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 {\left (b^2 (2 A b+5 a 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 {b^{3/2} (2 A b+5 a 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 {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}-\frac {\left ((a-i b)^3 (A-i 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 ((a+i b)^3 (A+i 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 {(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 {b^{3/2} (2 A b+5 a 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} (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 {b (2 a A+b B) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}}-\frac {2 a A \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.40 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.21 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {\cot (c+d x)} \left ((-1)^{3/4} (-a+i b)^{5/2} (i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-\sqrt [4]{-1} (a+i b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-(a-i b)^2 (A-i B) \sqrt {a+b \tan (c+d x)}-(a+i b)^2 (A+i B) \sqrt {a+b \tan (c+d x)}+b B (a+b \tan (c+d x))^{3/2}+\frac {b (2 A b+5 a B) \sqrt {a+b \tan (c+d x)} \left (\sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-\sqrt {a} \sqrt {1+\frac {b \tan (c+d x)}{a}}\right )}{\sqrt {a} \sqrt {1+\frac {b \tan (c+d x)}{a}}}\right )}{d} \]

[In]

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

[Out]

(Sqrt[Cot[c + d*x]]*((-1)^(3/4)*(-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]
])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - (-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]]]*Sqrt[Tan[c + d*x]] - (a - I*b)^2*(A - I*B)*Sqrt[a + b*
Tan[c + d*x]] - (a + I*b)^2*(A + I*B)*Sqrt[a + b*Tan[c + d*x]] + b*B*(a + b*Tan[c + d*x])^(3/2) + (b*(2*A*b +
5*a*B)*Sqrt[a + b*Tan[c + d*x]]*(Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - Sq
rt[a]*Sqrt[1 + (b*Tan[c + d*x])/a]))/(Sqrt[a]*Sqrt[1 + (b*Tan[c + d*x])/a])))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2667\) vs. \(2(251)=502\).

Time = 0.44 (sec) , antiderivative size = 2668, normalized size of antiderivative = 8.86

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2668\)
default	\(\text {Expression too large to display}\)	\(2668\)

[In]

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

[Out]

-1/4/d*((b+a*cot(d*x+c))/cot(d*x+c))^(1/2)/cot(d*x+c)^(1/2)*(4*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))*b^(7/2)*a*cot(d*x+c)-4*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))*b^(7/2)*a*cot(d*x+c)+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))*b^(5/2)*a^2*cot(d*x+c)-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))*b^(5/2)*a^2*cot(d*x+c)-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))*b^(3/2)*a^3*c
ot(d*x+c)+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))*
b^(3/2)*a^3*cot(d*x+c)-4*B*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(b+a*cot(d*x+c))^(1/2)*b^(5/2)*a-4*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))*b^(1/2)*a^4*cot(d*x+c)+4*B*arct
an((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))*b^(1/2)*a^4*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^(5/2)*a*cot(d*x+c)+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
^(5/2)*a*cot(d*x+c)-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^(5/2)*cot(d*x+c)-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)*b^(3/2)*a^2*cot(d*x+c)+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^(5
/2)*cot(d*x+c)+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)*b^(3/2)*a^2*cot(d*x+c)+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^(1/2)*a^3*cot(d*x+c)-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^(1/2)*a^3*cot(d*x+c)+4*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)*b^(1/2)*a
^3*cot(d*x+c)-4*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)*b^(1/2)*a^3*cot(d*x+c)+8*A*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*a^3*cot(d*x
+c)*b^(1/2)-8*A*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a*b^3*cot(d*x+c)-20*B*(2
*(a^2+b^2)^(1/2)-2*b)^(1/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a^2*b^2*cot(d*x+c)-4*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)*b^(5/2)*a*cot(d*x+
c)+4*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)*b^(5/2)*a*cot(d*x+c)-8*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)*b^(3/2)*a^2*cot(d*x+c)+8*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)*b^(3/2)*a^2*cot(d*x+c)+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^(7/2)*cot(d*x+c)-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^(7/2)*cot(d*x+c)+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^(1/2)*a^2*cot(d*x+c)-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^(1/2)*a^2*cot(d*x+c)+2*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^(3/2)*a*cot(d*x+c)-
2*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^(3/2)*a*cot(d*x+c))/a/(b+a*cot(d*x+c))^(1/2)/(2*(a
^2+b^2)^(1/2)-2*b)^(1/2)/b^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17670 vs. \(2 (245) = 490\).

Time = 7.58 (sec) , antiderivative size = 35373, normalized size of antiderivative = 117.52 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[In]

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

[Out]

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