\(\int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [589]

Optimal result

Integrand size = 33, antiderivative size = 325 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a (A-B)+b (A+B)) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3662, 3688, 3728, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} d \left (a^2+b^2\right )}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d} \]

[In]

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

[Out]

((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A
+ B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[a]
*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(a^2 + b^2)*d) + (2*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(a^2*d) - (2*A*Cot
[c + d*x]^(3/2))/(3*a*d) + ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqr
t[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*
(a^2 + b^2)*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 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 3662

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

Rule 3688

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3715

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

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*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] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{\frac {5}{2}}(c+d x) (B+A \cot (c+d x))}{b+a \cot (c+d x)} \, dx \\ & = -\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {2 \int \frac {\sqrt {\cot (c+d x)} \left (\frac {3 A b}{2}+\frac {3}{2} a A \cot (c+d x)+\frac {3}{2} (A b-a B) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{3 a} \\ & = \frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3}{4} b (A b-a B)-\frac {3}{4} a^2 B \cot (c+d x)-\frac {3}{4} \left (a^2 A-A b^2+a b B\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{3 a^2} \\ & = \frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3}{4} a^2 (A b-a B)-\frac {3}{4} a^2 (a A+b B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right )}+\frac {\left (b^3 (A b-a B)\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = \frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3}{4} a^2 (A b-a B)+\frac {3}{4} a^2 (a A+b B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {\left (2 b^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {(b (A-B)-a (A+B)) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \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} \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \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} \left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a (A-B)+b (A+B)) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (A+B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d} \\ & = \frac {(b (A-B)-a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 (A b-a B) \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a (A-B)+b (A+B)) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.84 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (-\frac {6 \sqrt {2} (b (-A+B)+a (A+B)) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {24 b^{5/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )}-\frac {3 \sqrt {2} (a (A-B)+b (A+B)) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{a^2+b^2}+\frac {8 A}{a \tan ^{\frac {3}{2}}(c+d x)}+\frac {24 (-A b+a B)}{a^2 \sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{12 d} \]

[In]

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

[Out]

-1/12*(Sqrt[Cot[c + d*x]]*((-6*Sqrt[2]*(b*(-A + B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcT
an[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*b^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x
]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) - (3*Sqrt[2]*(a*(A - B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] +
Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) + (8*A)/(a*Tan[c + d*x]^(3/2
)) + (24*(-(A*b) + a*B))/(a^2*Sqrt[Tan[c + d*x]]))*Sqrt[Tan[c + d*x]])/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(771\) vs. \(2(281)=562\).

Time = 0.50 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.38

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}+6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -3 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) a^{2} b +3 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{2} b +6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +3 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) a^{3}-24 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) b^{4}+24 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) a \,b^{3}-24 A \tan \left (d x +c \right ) \sqrt {a b}\, a^{2} b -24 A \tan \left (d x +c \right ) \sqrt {a b}\, b^{3}+24 B \tan \left (d x +c \right ) \sqrt {a b}\, a^{3}+24 B \tan \left (d x +c \right ) \sqrt {a b}\, a \,b^{2}+8 A \sqrt {a b}\, a^{3}+8 A \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(772\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}+6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-6 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -3 A \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) a^{2} b +3 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{2} b +6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+6 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +3 B \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) a^{3}-24 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) b^{4}+24 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) a \,b^{3}-24 A \tan \left (d x +c \right ) \sqrt {a b}\, a^{2} b -24 A \tan \left (d x +c \right ) \sqrt {a b}\, b^{3}+24 B \tan \left (d x +c \right ) \sqrt {a b}\, a^{3}+24 B \tan \left (d x +c \right ) \sqrt {a b}\, a \,b^{2}+8 A \sqrt {a b}\, a^{3}+8 A \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(772\)

[In]

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

[Out]

-1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(3*A*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1
/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^3+6*A*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(1
+2^(1/2)*tan(d*x+c)^(1/2))*a^3-6*A*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2
*b+6*A*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3-6*A*tan(d*x+c)^(3/2)*2^(1/
2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-3*A*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)
*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2*b+3*B*tan(d*x+c)^(3/2)*2^(1/2)*(a
*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^2*b+6*B*tan(d
*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+6*B*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2
)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+6*B*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+
c)^(1/2))*a^3+6*B*tan(d*x+c)^(3/2)*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+3*B*tan(d*x+c
)^(3/2)*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)
))*a^3-24*A*tan(d*x+c)^(3/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*b^4+24*B*tan(d*x+c)^(3/2)*arctan(b*tan(d*x
+c)^(1/2)/(a*b)^(1/2))*a*b^3-24*A*tan(d*x+c)*(a*b)^(1/2)*a^2*b-24*A*tan(d*x+c)*(a*b)^(1/2)*b^3+24*B*tan(d*x+c)
*(a*b)^(1/2)*a^3+24*B*tan(d*x+c)*(a*b)^(1/2)*a*b^2+8*A*(a*b)^(1/2)*a^3+8*A*(a*b)^(1/2)*a*b^2)/a^2/(a^2+b^2)/(a
*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3115 vs. \(2 (281) = 562\).

Time = 26.27 (sec) , antiderivative size = 6260, normalized size of antiderivative = 19.26 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {24 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{2} + b^{2}} - \frac {8 \, {\left (\frac {A a}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {3 \, {\left (B a - A b\right )}}{\sqrt {\tan \left (d x + c\right )}}\right )}}{a^{2}}}{12 \, d} \]

[In]

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

[Out]

1/12*(24*(B*a*b^3 - A*b^4)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4 + a^2*b^2)*sqrt(a*b)) + 3*(2*sqrt(2)
*((A + B)*a - (A - B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A + B)*a - (A - B)
*b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*((A - B)*a + (A + B)*b)*log(sqrt(2)/sqrt(t
an(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*((A - B)*a + (A + B)*b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d
*x + c) + 1))/(a^2 + b^2) - 8*(A*a/tan(d*x + c)^(3/2) + 3*(B*a - A*b)/sqrt(tan(d*x + c)))/a^2)/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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