\(\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\) [808]

Optimal result

Integrand size = 23, antiderivative size = 204 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]

[Out]

1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x
+c)^(1/2))/d*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a^2+2*a*b-b^
2)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-2*a^2*cot(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3754, 3624, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2 \sqrt {\cot (c+d x)}}{d} \]

[In]

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

[Out]

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

Rule 210

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3624

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

Rule 3754

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a \cot (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}+\int \frac {-a^2+b^2+2 a b \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {2 \text {Subst}\left (\int \frac {a^2-b^2-2 a b x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}+\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d} \\ & = -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.53 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)}+\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )+\frac {\left (a^2-b^2\right ) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{2 \sqrt {2}}}{d} \]

[In]

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

[Out]

-((2*a^2*Sqrt[Cot[c + d*x]] + (4*a*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 + (
(a^2 - b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqr
t[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*Sqrt[2]))/d)

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(-\frac {2 a^{2} \left (\sqrt {\cot }\left (d x +c \right )\right )+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}}{d}\)	\(201\)
default	\(-\frac {2 a^{2} \left (\sqrt {\cot }\left (d x +c \right )\right )+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}}{d}\)	\(201\)

[In]

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

[Out]

-1/d*(2*a^2*cot(d*x+c)^(1/2)+1/4*(-a^2+b^2)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)-
2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/2*a*b
*2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/
2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 991 vs. \(2 (174) = 348\).

Time = 0.29 (sec) , antiderivative size = 991, normalized size of antiderivative = 4.86 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {4 \, a^{2}}{\sqrt {\tan \left (d x + c\right )}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(d*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)*log(
((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)
*d)*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) + (a^8 -
4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(tan(d*x + c))) - d*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 -
12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)*log(-((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*
b^4 - 12*a^2*b^6 + b^8)/d^4) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 - 12*
a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) + (a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(ta
n(d*x + c))) - d*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/
d^2)*log(((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) + 2*(a^5*b - 6*a^3*b^3
+ a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)
+ (a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(tan(d*x + c))) + d*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt
(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)*log(-((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2
+ 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) + 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt(-(
a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) + (a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8
)*sqrt(tan(d*x + c))) + 4*a^2/sqrt(tan(d*x + c)))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.87 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, a^{2}}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

[In]

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

[Out]

1/4*(2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 - 2*a
*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)/sqrt
(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x
+ c) + 1) - 8*a^2/sqrt(tan(d*x + c)))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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