\(\int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx\) [818]

Optimal result

Integrand size = 23, antiderivative size = 272 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]

[Out]

8/5*a*b^2/d/cot(d*x+c)^(3/2)+2/5*b^2*(b+a*cot(d*x+c))/d/cot(d*x+c)^(5/2)-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2
^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/4*
(a-b)*(a^2+4*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-b)*(a^2+4*a*b+b^2)*ln(1+cot(d
*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+2*b*(3*a^2-b^2)/d/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3754, 3646, 3709, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2
)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (8*a*b^2)/(5*d*Cot[c + d*x]^(3/2)) + (2*b*(3*a^2 - b^2
))/(d*Sqrt[Cot[c + d*x]]) + (2*b^2*(b + a*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2)) + ((a - b)*(a^2 + 4*a*b + b^
2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sq
rt[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 3610

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

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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*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] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 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))^3}{\cot ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int \frac {-6 a b^2-\frac {5}{2} b \left (3 a^2-b^2\right ) \cot (c+d x)-\frac {1}{2} a \left (5 a^2-3 b^2\right ) \cot ^2(c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int \frac {-\frac {5}{2} b \left (3 a^2-b^2\right )-\frac {5}{2} a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int \frac {-\frac {5}{2} a \left (a^2-3 b^2\right )+\frac {5}{2} b \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 \text {Subst}\left (\int \frac {\frac {5}{2} a \left (a^2-3 b^2\right )-\frac {5}{2} b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{5 d} \\ & = \frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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+b) \left (a^2-4 a b+b^2\right )\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-b) \left (a^2+4 a b+b^2\right )\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-b) \left (a^2+4 a b+b^2\right )\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 {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {(a-b) \left (a^2+4 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 {(a-b) \left (a^2+4 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+b) \left (a^2-4 a b+b^2\right )\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+b) \left (a^2-4 a b+b^2\right )\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 {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {(a-b) \left (a^2+4 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 {(a-b) \left (a^2+4 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 = 0.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.38 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 \left (\left (-9 a^2 b+3 b^3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )+a \left (a (9 b+5 a \cot (c+d x))-5 \left (a^2-3 b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{15 d \cot ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

(2*((-9*a^2*b + 3*b^3)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + a*(a*(9*b + 5*a*Cot[c + d*x]) - 5*(
a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*Cot[c + d*x]^(5/2))

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(246\)
default	\(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(246\)

[In]

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

[Out]

-1/d*(-2/5*b^3/cot(d*x+c)^(5/2)-2*b*(3*a^2-b^2)/cot(d*x+c)^(1/2)-2*a*b^2/cot(d*x+c)^(3/2)+1/4*(a^3-3*a*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)))+1/4*(-3*a^2*b+b^3)*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. 1399 vs. \(2 (234) = 468\).

Time = 0.36 (sec) , antiderivative size = 1399, normalized size of antiderivative = 5.14 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/10*(5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 +
255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4
- 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9
)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^
4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*s
qrt(tan(d*x + c))) - 5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 -
452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2
+ 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 1
8*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a
^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2
*b^10 - b^12)*sqrt(tan(d*x + c))) - 5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a^10*b^2 +
255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12
- 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (3*a^8*b - 46*a^6*b^3 +
60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a
^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^
4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) + 5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 -
30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((a^3 - 3*a*b^2)*
d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (3*a^8*b
- 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a
^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*
a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) - 4*(b^3*tan(d*x + c)^3 + 5*a*b^2*tan(d*x + c)^
2 + 5*(3*a^2*b - b^3)*tan(d*x + c))/sqrt(tan(d*x + c)))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {8 \, {\left (b^{3} + \frac {5 \, a b^{2}}{\tan \left (d x + c\right )} + \frac {5 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{20 \, d} \]

[In]

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

[Out]

1/20*(8*(b^3 + 5*a*b^2/tan(d*x + c) + 5*(3*a^2*b - b^3)/tan(d*x + c)^2)*tan(d*x + c)^(5/2) - 10*sqrt(2)*(a^3 -
3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) - 10*sqrt(2)*(a^3 - 3*a^2*b - 3
*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3
)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2
)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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