3.7.35 \(\int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) [635]

Optimal. Leaf size=457 \[ -\frac {(i a-b)^{5/2} (i A-B) \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}} \]

[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+1/64*(40*A*a^3*b-320*A*a*b^3-5*B*a^4-240*B*a^2*b^2+128*B*b^4)*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)/b^(3/2)/d-(I*a+b)^(5/2)*(I*A+B)*arctanh((I*a+b)^(1/2)*ta
n(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+1/64*(40*A*a^2*b-64*A*b^3-5*B*a^3-1
12*B*a*b^2)*(a+b*tan(d*x+c))^(1/2)/b/d/cot(d*x+c)^(1/2)+1/96*(40*A*a*b-5*B*a^2-48*B*b^2)*(a+b*tan(d*x+c))^(3/2
)/b/d/cot(d*x+c)^(1/2)+1/24*(8*A*b-B*a)*(a+b*tan(d*x+c))^(5/2)/b/d/cot(d*x+c)^(1/2)+1/4*B*(a+b*tan(d*x+c))^(7/
2)/b/d/cot(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.24, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {4326, 3688, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \begin {gather*} \frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (-5 a^4 B+40 a^3 A b-240 a^2 b^2 B-320 a A b^3+128 b^4 B\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {ArcTan}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),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) + ((40*a^3*A*b - 320*a*A*b^3 - 5*a^4*B - 240*a^2*b^2*B + 128*b^4*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]])/(64*b^(3/2)*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 + ((40*a^2*A*b - 64*A*b^3 - 5*a^3*B - 112*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(64*b*d*Sq
rt[Cot[c + d*x]]) + ((40*a*A*b - 5*a^2*B - 48*b^2*B)*(a + b*Tan[c + d*x])^(3/2))/(96*b*d*Sqrt[Cot[c + d*x]]) +
((8*A*b - a*B)*(a + b*Tan[c + d*x])^(5/2))/(24*b*d*Sqrt[Cot[c + d*x]]) + (B*(a + b*Tan[c + d*x])^(7/2))/(4*b*
d*Sqrt[Cot[c + d*x]])

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 3688

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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*} \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\\ &=\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2} \left (-\frac {a B}{2}-4 b B \tan (c+d x)+\frac {1}{2} (8 A b-a B) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{4 b}\\ &=\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\frac {1}{4} a (8 A b+5 a B)-12 b (A b+a B) \tan (c+d x)+\frac {1}{4} \left (40 a A b-5 a^2 B-48 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{12 b}\\ &=\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {3}{8} a \left (24 a A b+5 a^2 B-16 b^2 B\right )-24 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{8} \left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {3}{16} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {-\frac {3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x+\frac {3}{16} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {3 \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right )}{16 \sqrt {x} \sqrt {a+b x}}-\frac {24 \left (b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+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 )}{b d}+\frac {\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \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 )}{128 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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 {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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 )}{b d}+\frac {\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \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 )}{64 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left ((i a+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 (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \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 )}{64 b d}-\frac {\left (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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 b d}\\ &=\frac {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\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)}}{64 b^{3/2} d}+\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left ((i a+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 (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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 )}{b d}\\ &=-\frac {(i a-b)^{5/2} (i A-B) \tan ^{-1}\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 {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\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)}}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} (i A+B) \tanh ^{-1}\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 {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 3.66, size = 431, normalized size = 0.94 \begin {gather*} \frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-192 \sqrt [4]{-1} (-a+i b)^{5/2} b (i A+B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+192 (-1)^{3/4} (a+i b)^{5/2} b (A+i B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-3 \left (-40 a^2 A b+64 A b^3+5 a^3 B+112 a b^2 B\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}-2 \left (-40 a A b+5 a^2 B+48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 (8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}+48 B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}-\frac {3 \sqrt {a} \left (-40 a^3 A b+320 a A b^3+5 a^4 B+240 a^2 b^2 B-128 b^4 B\right ) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}\right )}{192 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-192*(-1)^(1/4)*(-a + I*b)^(5/2)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-
a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 192*(-1)^(3/4)*(a + I*b)^(5/2)*b*(A + I*B)*ArcTan[((-
1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-40*a^2*A*b + 64*A*b^3 + 5*a^3*B + 1
12*a*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] - 2*(-40*a*A*b + 5*a^2*B + 48*b^2*B)*Sqrt[Tan[c + d*x]
]*(a + b*Tan[c + d*x])^(3/2) + 8*(8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) + 48*B*Sqrt[Tan[c
+ d*x]]*(a + b*Tan[c + d*x])^(7/2) - (3*Sqrt[a]*(-40*a^3*A*b + 320*a*A*b^3 + 5*a^4*B + 240*a^2*b^2*B - 128*b^
4*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d
*x]])))/(192*b*d)

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 31.10, size = 39827, normalized size = 87.15

method	result	size
default	\(\text {Expression too large to display}\)	\(39827\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4371 deep

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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