\(\int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [862]

Optimal result

Integrand size = 25, antiderivative size = 187 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d}-\frac {2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4326, 3650, 12, 3656, 924, 95, 211, 214} \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {i \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {i \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}-\frac {2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{a d} \]

[In]

Int[Cot[c + d*x]^(3/2)/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((-I)*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) + (I*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) - (2*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(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 924

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

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}-\frac {2 \sqrt {a+b \tan (c+d x)}}{a \sqrt {\tan (c+d x)}}\right )}{d} \]

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1495\) vs. \(2(153)=306\).

Time = 40.49 (sec) , antiderivative size = 1496, normalized size of antiderivative = 8.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1496\)

[In]

int(cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4/d*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c)))^(3/2)*((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)*(ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-co
s(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*(c
sc(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*(
b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))-(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2)
)^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*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-co
s(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))*(csc(d*x+c)-cot(d*x+c))-ln(1/(1-cos(d*
x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+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*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d
*x+c))-sin(d*x+c)*a))*b*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(a^2+b^2)
^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*ln(1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2
+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*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(csc(d*x+c)-co
t(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*(csc(d*x+c)-cot(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*(csc(d*x+c)-cot(d*x+c))+4*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(cs
c(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)^2*(1-cos
(d*x+c))^2-1)/(-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)*(csc(d*x+c)-cot(d*x+c))*2^(1/2)/a/(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. 4115 vs. \(2 (147) = 294\).

Time = 0.68 (sec) , antiderivative size = 4115, normalized size of antiderivative = 22.01 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/8*(a*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((((a^6 +
7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4
*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d
^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)*ta
n(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)) + a*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(-(((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3
- 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x +
c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)) - a*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((((a^6 + 7*a^4*b^2 + 1
2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d +
2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x +
c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*d^3)*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*d^2*sqr
t(-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)) - a*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(-(((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d
*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 +
6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)) + a*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((((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*t
an(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d - 2*((a^4*b^3 +
5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b +
3*a^4*b^3 + 2*a^2*b^5)*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)) + a*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(-(((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) -
(a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*
a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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*ta
n(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)) - a*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((((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2
*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^
7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2
*b^5)*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)) - a*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(-(((a^6
+ 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2
+ 4*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6
)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)) - 16*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(a*d)

Sympy [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]

[In]

integrate(cot(d*x+c)**(3/2)/(a+b*tan(d*x+c))**(1/2),x)

[Out]

Integral(cot(c + d*x)**(3/2)/sqrt(a + b*tan(c + d*x)), x)

Maxima [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \]

[In]

int(cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(1/2), x)