\(\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [849]

Optimal result

Integrand size = 25, antiderivative size = 185 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4326, 3648, 3697, 3696, 95, 209, 212} \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (-b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

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

[Out]

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

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 3648

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

Rule 3696

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]

Rule 3697

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 -
I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e +
f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2
+ B^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 {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-a b+\frac {1}{2} \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (i (a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} \left (i (a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}+\frac {\left (i (a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left (i (a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {2 a \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}+\frac {\left (i (a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (i (a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \\ & = \frac {i (i a-b)^{3/2} \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 {i (i a+b)^{3/2} \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 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-\sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-2 a \sqrt {a+b \tan (c+d x)}\right )}{d} \]

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2229\) vs. \(2(151)=302\).

Time = 37.05 (sec) , antiderivative size = 2230, normalized size of antiderivative = 12.05

method	result	size
default	\(\text {Expression too large to display}\)	\(2230\)

[In]

int(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(3/2),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-c
os(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)*(a^2+b^2)^(1/2)*b*(-b+(a^2+b^2)^(1/2))^(1/2)*(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*(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)*(a^2+b^2)^(1/2)*b*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*
x+c))+a^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)*(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*(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^2*(-b+(a^2+b^2)^(1/2))^(1/2)*(c
sc(d*x+c)-cot(d*x+c))-a^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)*(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*(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^2*(-b+(a^2+b^
2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+2*a^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))*s
in(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*(a^2+b^2)^(1/2)*(csc(d*x+c)-cot(d*x+c))+2*a^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)*(csc(d*x+c)-cot(d
*x+c))-4*a^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))*b*(csc(d*x+c)-cot(d*x+c))-4*a^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))*b*(csc(d*x+c)-cot(d*x+c))-4*a^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)/(-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/(-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. 3545 vs. \(2 (145) = 290\).

Time = 0.58 (sec) , antiderivative size = 3545, normalized size of antiderivative = 19.16 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((((a^6 + a^4*b^2 - 12*a^2*
b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d -
2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^
4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a
*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2
- (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) +
a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)
/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*
x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*
x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b
^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c)
+ (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^
2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - d*sqrt((3*a^2*b - b^
3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2
*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c)
+ 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b -
b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*
(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^
4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(t
an(d*x + c)^2 + 1)) - d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 +
a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2
+ 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*
a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7
- 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d
^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt
(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^
4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 1
2*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b +
4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*
a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*
b^5)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt
(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + d*s
qrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*
tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3
*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4)
)*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*t
an(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4
+ 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt
(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/
d^2)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) -
(a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2
)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4)
)/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a
^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^
2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - d*sqrt((3*a^2*b - b^3 - d^2*
sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5
*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2
*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d
^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b
- a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*
tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^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)*a/sqrt(tan(d*x + c)))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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