\(\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [633]
Optimal result
Integrand size = 35, antiderivative size = 320 \[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(i a-b)^{5/2} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\sqrt {b} \left (20 a A b+15 a^2 B-8 b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{4 d}+\frac {(i a+b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}
\]
[Out]
(I*a-b)^(5/2)*(I*A-B)*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)^(5/2)*(I*A+B)*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+1/4*(20*A*a*b+15*B*a^2-8*B*b^2)*arctanh(b^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*
b^(1/2)*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+1/4*b*(4*A*b+7*B*a)*(a+b*tan(d*x+c))^(1/2)/d/cot(d*x+c)^(1/2)+1/2*
b*B*(a+b*tan(d*x+c))^(3/2)/d/cot(d*x+c)^(1/2)
Rubi [A] (verified)
Time = 2.65 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of
steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {4326, 3688, 3728, 3736,
6857, 65, 223, 212, 95, 211, 214} \[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {b} \left (15 a^2 B+20 a A b-8 b^2 B\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{4 d}+\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {b (7 a B+4 A b) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}
\]
[In]
Int[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
((I*a - b)^(5/2)*(I*A - B)*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 + (Sqrt[b]*(20*a*A*b + 15*a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt
[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d) + ((I*a + b)^(5/2)*(I*A + B)*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]])/d + (b*(4*A*b +
7*a*B)*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]) + (b*B*(a + b*Tan[c + d*x])^(3/2))/(2*d*Sqrt[Cot[c +
d*x]])
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 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 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 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 223
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
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 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 3736
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2)), x], x, Tan[
e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, 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]
Rule 6857
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
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))^{5/2} (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {1}{2} \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {1}{2} a (4 a A-b B)+2 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {1}{2} b (4 A b+7 a B) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {1}{2} \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} a \left (8 a^2 A-4 A b^2-9 a b B\right )+2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {1}{4} b \left (20 a A b+15 a^2 B-8 b^2 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} a \left (8 a^2 A-4 A b^2-9 a b B\right )+2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x+\frac {1}{4} b \left (20 a A b+15 a^2 B-8 b^2 B\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {b \left (20 a A b+15 a^2 B-8 b^2 B\right )}{4 \sqrt {x} \sqrt {a+b x}}+\frac {2 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {a^3 A-3 a A b^2-3 a^2 b B+b^3 B+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (b \left (20 a A b+15 a^2 B-8 b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {-3 a^2 A b+A b^3-a^3 B+3 a b^2 B+i \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {3 a^2 A b-A b^3+a^3 B-3 a b^2 B+i \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (b \left (20 a A b+15 a^2 B-8 b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 d} \\ & = \frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left ((a-i b)^3 (i A+B) \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 {\left (b \left (20 a A b+15 a^2 B-8 b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{4 d}+\frac {\left (\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B+i \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \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 {\sqrt {b} \left (20 a A b+15 a^2 B-8 b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{4 d}+\frac {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {\left ((a-i b)^3 (i A+B) \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 (\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B+i \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \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 a-b)^{5/2} (i A-B) \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 {\sqrt {b} \left (20 a A b+15 a^2 B-8 b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{4 d}+\frac {(i a+b)^{5/2} (i A+B) \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 {b (4 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.77 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.97
\[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (4 \sqrt [4]{-1} (-a+i b)^{5/2} (i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-4 (-1)^{3/4} (a+i b)^{5/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+b (4 A b+7 a B) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}+2 b B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+\frac {\sqrt {a} \sqrt {b} \left (20 a A b+15 a^2 B-8 b^2 B\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {a+b \tan (c+d x)}}\right )}{4 d}
\]
[In]
Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(4*(-1)^(1/4)*(-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I
*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 4*(-1)^(3/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*
Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*(4*A*b + 7*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*
Tan[c + d*x]] + 2*b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + (Sqrt[a]*Sqrt[b]*(20*a*A*b + 15*a^2*B -
8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])
)/(4*d)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2818\) vs. \(2(264)=528\).
Time = 0.51 (sec) , antiderivative size = 2819, normalized size of antiderivative =
8.81
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method | result | size |
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derivativedivides |
\(\text {Expression too large to display}\) |
\(2819\) |
default |
\(\text {Expression too large to display}\) |
\(2819\) |
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[In]
int(cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/4/d*((b+a*cot(d*x+c))/cot(d*x+c))^(1/2)/cot(d*x+c)^(3/2)*(2*B*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(b+a*cot(d*x+c))
^(1/2)*b^(5/2)*a-8*A*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(
1/2))*(a^2+b^2)^(1/2)*b^(3/2)*a^2*cot(d*x+c)^2-4*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1
/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*b^(5/2)*a*cot(d*x+c)^2+4*B*arctan((2*(b+a*cot(d*x+c))^(1/2
)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*b^(5/2)*a*cot(d*x+c)^2-A*ln((b
+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/
2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^(7/2)*cot(d*x+c)^2+A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^
(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^(7/2)*cot(d*x+
c)^2+8*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b
^2)^(1/2)*b^(3/2)*a^2*cot(d*x+c)^2+4*A*a*b^(5/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(b+a*cot(d*x+c))^(1/2)*cot(d*x+
c)+9*B*a^2*b^(3/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(b+a*cot(d*x+c))^(1/2)*cot(d*x+c)+20*A*(2*(a^2+b^2)^(1/2)-2*b
)^(1/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a^2*b^2*cot(d*x+c)^2+4*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)
-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*b^(1/2)*a^3*cot(d*x+c)^2-4*B*arctan(
(2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*(a^2+b^2)^(1/2)*b^(1/2
)*a^3*cot(d*x+c)^2+15*B*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a^3*b*cot(d*x+c)
^2-8*B*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*arctanh((b+a*cot(d*x+c))^(1/2)/b^(1/2))*a*b^3*cot(d*x+c)^2+B*ln((b+a*cot(
d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(
a^2+b^2)^(1/2)-2*b)^(1/2)*b^(1/2)*a^3*cot(d*x+c)^2-B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/
2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^(1/2)*a^3*cot(d*x
+c)^2+A*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(
1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^(5/2)*cot(d*x+c)^2+3*A*ln((b+a*cot(d*x+c))^(1/
2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1
/2)-2*b)^(1/2)*b^(3/2)*a^2*cot(d*x+c)^2-A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/
2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^(5/2)*cot(d*
x+c)^2-3*A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2
)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^(3/2)*a^2*cot(d*x+c)^2-3*B*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^
2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^
(1/2)*b^(5/2)*a*cot(d*x+c)^2+3*B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b
^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*b^(5/2)*a*cot(d*x+c)^2-A*ln((b+a*cot(d*
x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^
2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^(1/2)*a^2*cot(d*x+c)^2+A*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2
*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+
b^2)^(1/2)*b^(1/2)*a^2*cot(d*x+c)^2+2*B*ln((b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)-a*cot(d*x+c)-b
-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^(3/2)*a*cot(d*
x+c)^2-2*B*ln(a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2
)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*(a^2+b^2)^(1/2)*b^(3/2)*a*cot(d*x+c)^2+4*A*arctan(((2*(a^2+b^
2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(1/2)*a^4*cot(d*x+c)^2-4*A*arct
an((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(1/2)*a^4*cot(d*x
+c)^2-12*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(3
/2)*a^3*cot(d*x+c)^2+12*B*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2
*b)^(1/2))*b^(3/2)*a^3*cot(d*x+c)^2+4*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*(a^
2+b^2)^(1/2)-2*b)^(1/2))*b^(7/2)*a*cot(d*x+c)^2-4*B*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(
1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(7/2)*a*cot(d*x+c)^2-12*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*
cot(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(5/2)*a^2*cot(d*x+c)^2+12*A*arctan((2*(b+a*cot(d*x+c))^(1/
2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*b^(5/2)*a^2*cot(d*x+c)^2)/a/(b+a*cot(d*x+c))^
(1/2)/(2*(a^2+b^2)^(1/2)-2*b)^(1/2)/b^(1/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 17723 vs. \(2 (258) = 516\).
Time = 7.67 (sec) , antiderivative size = 35483, normalized size of antiderivative = 110.88
\[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F(-1)]
Timed out. \[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out}
\]
[In]
integrate(cot(d*x+c)**(1/2)*(a+b*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
[Out]
Timed out
Maxima [F]
\[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cot \left (d x + c\right )} \,d x }
\]
[In]
integrate(cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^(5/2)*sqrt(cot(d*x + c)), x)
Giac [F(-1)]
Timed out. \[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out}
\]
[In]
integrate(cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int \sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x
\]
[In]
int(cot(c + d*x)^(1/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2),x)
[Out]
int(cot(c + d*x)^(1/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2), x)