\(\int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [861]

Optimal result

Integrand size = 25, antiderivative size = 220 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}-\frac {\text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d}+\frac {4 b \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3 a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 a d} \]

[Out]

-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)^(1/
2)-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/(I*a+b)^
(1/2)-2/3*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/a/d+4/3*b*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1/2)/a^2/d

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4326, 3650, 3730, 12, 3656, 926, 95, 211, 214} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {4 b \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3 a^2 d}-\frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}-\frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 a d} \]

[In]

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

[Out]

-((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]])/
(Sqrt[I*a - b]*d)) - (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]])/(Sqrt[I*a + b]*d) + (4*b*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*a^2*d) - (2*Cot[c
+ d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*a*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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 3656

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Wit
h[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*((c + d*ff*x)^n/(1 + ff^2*x^2)), x]
, x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

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

Mathematica [A] (verified)

Time = 2.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {3 (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {3 (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}-\frac {2 (a-2 b \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}{a^2 \tan ^{\frac {3}{2}}(c+d x)}\right )}{3 d} \]

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1636\) vs. \(2(180)=360\).

Time = 41.82 (sec) , antiderivative size = 1637, normalized size of antiderivative = 7.44

method	result	size
default	\(\text {Expression too large to display}\)	\(1637\)

[In]

int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c)))^(5/2)*(1-cos(d*x+c))*((csc(d*x+
c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(3*csc(d*x+c)^
2*ln(-1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*sin(d*x+c)*(-csc(d*x
+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^
(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a^2*(1-cos(d*x+c)
)^2-3*csc(d*x+c)^2*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(
d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(
a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a
^2*(1-cos(d*x+c))^2-6*csc(d*x+c)^2*arctan(((b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc
(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(
-b+(a^2+b^2)^(1/2))^(1/2))*a^2*b*(1-cos(d*x+c))^2+6*csc(d*x+c)^2*arctan(((b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)
-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2
))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*(a^2+b^2)^(1/2)*a^2*(1-cos(d*x+c))^2-6*csc(d*x+c)^2*a
rctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(c
sc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2*(1-c
os(d*x+c))^2*b+6*csc(d*x+c)^2*arctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x
+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(
a^2+b^2)^(1/2))^(1/2))*a^2*(1-cos(d*x+c))^2*(a^2+b^2)^(1/2)+2*csc(d*x+c)^2*a*(a^2+b^2)^(1/2)*(-csc(d*x+c)*(csc
(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(
1-cos(d*x+c))^2+8*b*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)
))^(1/2)*(a^2+b^2)^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))-2*a*(a^2+b^2)^(1/2)*(-csc(d*x+c)*(
csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2
))/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1)^2/(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c
))-a)*(1-cos(d*x+c)))^(1/2)*2^(1/2)/a^2/(a^2+b^2)^(1/2)/(-b+(a^2+b^2)^(1/2))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4220 vs. \(2 (176) = 352\).

Time = 0.74 (sec) , antiderivative size = 4220, normalized size of antiderivative = 19.18 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-1/24*(3*a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2))*log(1/2*
((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b
^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b
^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*
sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2)) + 2*((a^5 + 3*a^3*b^2 +
4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d*x + c) - (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x +
c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt
(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) + 3*a^2*d*sqrt(((a^2 + b^2)*d^2*sq
rt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)
^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4
+ 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^
2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2)) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b +
2*a^2*b^3)*tan(d*x + c) - (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 +
4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)
))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 3*a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))
+ b)/((a^2 + b^2)*d^2))*log(1/2*((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*
tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^
6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2
/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b
^2)*d^2)) - 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d*x + c) - (2*(a^5*b + 3
*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) -
3*a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(
a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d
- ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d
^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(
((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2)) - 2*((a^5 + 3*a^3*b^2 + 4*a*
b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d*x + c) - (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2
- (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*ta
n(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) + 3*a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-
a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2))*log(1/2*((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 +
2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12
*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7
*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2)) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^
2*b^3)*tan(d*x + c) + (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b
^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(
tan(d*x + c)^2 + 1))*tan(d*x + c) + 3*a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) -
b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*ta
n(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*
b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/(
(a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^
2)*d^2)) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d*x + c) + (2*(a^5*b + 3*
a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^
4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) -
3*a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2))*log(1/2*((2*(a
^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 + 2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d
+ ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^
3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-
((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2)) - 2*((a^5 + 3*a^3*b^2 + 4*a*
b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d*x + c) + (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2
- (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*ta
n(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 3*a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-
a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(a^3*b^3 + 4*a*b^5)*d*tan(d*x + c)^2 +
2*(a^6 + 5*a^4*b^2 + 8*a^2*b^4)*d*tan(d*x + c) + 2*(a^5*b + 2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 1
2*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 +
7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4)) - b)/((a^2 + b^2)*d^2)) - 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a
^2*b^3)*tan(d*x + c) + (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a^4*b^2 + 7*a^2*b^4 + 4*
b^6)*d^2*tan(d*x + c))*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/
(tan(d*x + c)^2 + 1))*tan(d*x + c) - 16*(2*b*tan(d*x + c) - a)*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(a
^2*d*tan(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cot(d*x+c)**(5/2)/(a+b*tan(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [F(-1)]

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

[In]

int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(1/2), x)