\(\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx\) [98]
Optimal result
Integrand size = 27, antiderivative size = 151 \[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=-\frac {(i a-b) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}
\]
[Out]
-(I*a-b)*(a-I*b)^(5/2)*arctanh((a+b*cot(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(5/2)*(I*a+b)*arctanh((a+b*cot(
d*x+c))^(1/2)/(a+I*b)^(1/2))/d-2/5*b*(a+b*cot(d*x+c))^(5/2)/d+2*b*(a^2+b^2)*(a+b*cot(d*x+c))^(1/2)/d
Rubi [A] (verified)
Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of
steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3609, 12, 3563, 3620, 3618,
65, 214} \[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {(-b+i a) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (b+i a) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}
\]
[In]
Int[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2),x]
[Out]
-(((I*a - b)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d) + ((a + I*b)^(5/2)*(I*a + b)*
ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(a^2 + b^2)*Sqrt[a + b*Cot[c + d*x]])/d - (2*b*(a +
b*Cot[c + d*x])^(5/2))/(5*d)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 3563
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 + b^2, 0] && GtQ[n, 1]
Rule 3609
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Rule 3618
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Rule 3620
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rubi steps \begin{align*}
\text {integral}& = -\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}+\int \left (-a^2-b^2\right ) (a+b \cot (c+d x))^{3/2} \, dx \\ & = -\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}+\left (-a^2-b^2\right ) \int (a+b \cot (c+d x))^{3/2} \, dx \\ & = \frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}+\left (-a^2-b^2\right ) \int \frac {a^2-b^2+2 a b \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx \\ & = \frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}-\frac {1}{2} \left ((a-i b)^2 \left (a^2+b^2\right )\right ) \int \frac {1+i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx-\frac {1}{2} \left ((a+i b)^2 \left (a^2+b^2\right )\right ) \int \frac {1-i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx \\ & = \frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}-\frac {\left ((a+i b)^3 (i a+b)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \cot (c+d x)\right )}{2 d}-\frac {\left ((a+i b) (i a+b)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \cot (c+d x)\right )}{2 d} \\ & = \frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d}-\frac {\left ((a-i b)^3 (a+i b)\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{b d}-\frac {\left ((a-i b) (a+i b)^3\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{b d} \\ & = -\frac {(i a-b) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{5/2}}{5 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.47
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\frac {b \left (\frac {5 \left (a^2+b^2\right ) \left (a^2-b^2-2 a \sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}-\frac {5 \left (a^2+b^2\right ) \left (a^2-b^2+2 a \sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}+10 \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}-2 (a+b \cot (c+d x))^{5/2}\right )}{5 d}
\]
[In]
Integrate[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2),x]
[Out]
(b*((5*(a^2 + b^2)*(a^2 - b^2 - 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[
-b^2]*Sqrt[a - Sqrt[-b^2]]) - (5*(a^2 + b^2)*(a^2 - b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqr
t[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + 10*(a^2 + b^2)*Sqrt[a + b*Cot[c + d*x]] - 2*(a + b*Cot
[c + d*x])^(5/2)))/(5*d)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1374\) vs. \(2(127)=254\).
Time = 0.08 (sec) , antiderivative size = 1375, normalized size of antiderivative =
9.11
| | |
method | result | size |
| | |
derivativedivides |
\(\text {Expression too large to display}\) |
\(1375\) |
default |
\(\text {Expression too large to display}\) |
\(1375\) |
parts |
\(\text {Expression too large to display}\) |
\(2386\) |
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[In]
int((-a+b*cot(d*x+c))*(a+b*cot(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
[Out]
-2/5*b*(a+b*cot(d*x+c))^(5/2)/d+2/d*b*(a+b*cot(d*x+c))^(1/2)*a^2+2/d*b^3*(a+b*cot(d*x+c))^(1/2)+1/4/d/b*ln((a+
b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2
)*(a^2+b^2)^(1/2)*a^3+1/4/d*b*ln((a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(
1/2)-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-1/4/d/b*ln((a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2
*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+1/4/d*b^3*ln((a+b*cot(d*x+c))^(1/2
)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*cot(d*x+c)-(a^2+b^2)^(1/2)-a)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(
1/2))*(a^2+b^2)^(1/2)*a^2+1/d*b^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)
^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*(a^2+b^2)^(1/2)-2/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((
-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3-2/d*b^3/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(
1/2))*a-1/4/d/b*ln(b*cot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^
2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3-1/4/d*b*ln(b*cot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2
)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a+1/4/d/b*ln(b*cot(d*x+c)+a+(a+b*c
ot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4-1/4/d*b^3*ln
(b*cot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^
(1/2)-1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(
a^2+b^2)^(1/2)-2*a)^(1/2))*(a^2+b^2)^(1/2)*a^2-1/d*b^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c)
)^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*(a^2+b^2)^(1/2)+2/d*b/(2*(a^2+b^2)^(1/2)
-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3
+2/d*b^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2))*a
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1684 vs. \(2 (118) = 236\).
Time = 0.31 (sec) , antiderivative size = 1684, normalized size of antiderivative = 11.15
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((-a+b*cot(d*x+c))*(a+b*cot(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
1/10*(5*(d*cos(2*d*x + 2*c) - d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*
b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6
*b^5 + 6*a^4*b^7 - a^2*b^9 - b^11)*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) + (a*d
^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) + (3*a^6*b^
2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10
*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)) - 5*(d*cos(2*d*x + 2*c) - d)*sqrt(
-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b
^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4*b^7 - a^2*b^9 - b^11)*sqr
t((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) - (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 3
1*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) + (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt
(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*
b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)) - 5*(d*cos(2*d*x + 2*c) - d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6
- d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*lo
g(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4*b^7 - a^2*b^9 - b^11)*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x +
2*c) + b)/sin(2*d*x + 2*c)) + (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2
*a^2*b^12 + b^14)/d^4) - (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6
- d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2))
+ 5*(d*cos(2*d*x + 2*c) - d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4
+ 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5
+ 6*a^4*b^7 - a^2*b^9 - b^11)*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) - (a*d^3*s
qrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) - (3*a^6*b^2 +
5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4
+ 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)) + 8*(a*b^2*sin(2*d*x + 2*c) - 2*a^2*b
- 2*b^3 + (2*a^2*b + 3*b^3)*cos(2*d*x + 2*c))*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2
*c)))/(d*cos(2*d*x + 2*c) - d)
Sympy [F]
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=- \int a^{3} \sqrt {a + b \cot {\left (c + d x \right )}}\, dx - \int \left (- b^{3} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\right )\, dx - \int \left (- a b^{2} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\right )\, dx - \int a^{2} b \sqrt {a + b \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx
\]
[In]
integrate((-a+b*cot(d*x+c))*(a+b*cot(d*x+c))**(5/2),x)
[Out]
-Integral(a**3*sqrt(a + b*cot(c + d*x)), x) - Integral(-b**3*sqrt(a + b*cot(c + d*x))*cot(c + d*x)**3, x) - In
tegral(-a*b**2*sqrt(a + b*cot(c + d*x))*cot(c + d*x)**2, x) - Integral(a**2*b*sqrt(a + b*cot(c + d*x))*cot(c +
d*x), x)
Maxima [F]
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (b \cot \left (d x + c\right ) - a\right )} \,d x }
\]
[In]
integrate((-a+b*cot(d*x+c))*(a+b*cot(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
integrate((b*cot(d*x + c) + a)^(5/2)*(b*cot(d*x + c) - a), x)
Giac [F]
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (b \cot \left (d x + c\right ) - a\right )} \,d x }
\]
[In]
integrate((-a+b*cot(d*x+c))*(a+b*cot(d*x+c))^(5/2),x, algorithm="giac")
[Out]
integrate((b*cot(d*x + c) + a)^(5/2)*(b*cot(d*x + c) - a), x)
Mupad [B] (verification not implemented)
Time = 40.09 (sec) , antiderivative size = 3442, normalized size of antiderivative = 22.79
\[
\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx=\text {Too large to display}
\]
[In]
int(-(a + b*cot(c + d*x))^(5/2)*(a - b*cot(c + d*x)),x)
[Out]
log(((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2
)*(((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)
*(64*a^2*b^5 + 64*a^4*b^3 - 32*a*b^2*d*(((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b
^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) - (16*a^2*b^2*(a + b*cot(c + d*x))^(1/2
)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2))/2 - (8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3)*((20*a^6*b^8*d
^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^10*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2)/(4*d^4) - a^7/(4*d^2) - (5*a^3
*b^4)/(4*d^2) + (5*a^5*b^2)/(2*d^2))^(1/2) - log(((-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d
^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(((-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d
^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 + 32*a*b^2*d*(-((-a^4*b^2*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)
))/(2*d) + (16*a^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2))/2 - (8*a^3*b^3*
(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3)*(-(a^7*d^2 + (20*a^6*b^8*d^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^10*b^4
*d^4 - 25*a^12*b^2*d^4)^(1/2) + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/(4*d^4))^(1/2) + log(((-((-a^4*b^2*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(((-((-a^4*b^2*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 -
32*a*b^2*d*(-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d
^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) - (16*a^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4
- 15*a^4*b^2))/d^2))/2 - (8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3)*((5*a^5*b^2)/(2*d^2) - a^7/(4*d^2) - (5*
a^3*b^4)/(4*d^2) - (20*a^6*b^8*d^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^10*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2
)/(4*d^4))^(1/2) - ((4*a^2*b)/d - (2*b*(a^2 + b^2))/d)*(a + b*cot(c + d*x))^(1/2) - log(((((((-b^6*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2)*(32*b^7 - 32*a^4*b^3 + 3
2*a*b^2*d*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(
1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) + (16*(a + b*cot(c + d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6
*b^4))/d^2)*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)
^(1/2))/2 + (8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*(((20*a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^10*d^4 + 100*
a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/(4*d^4))^(1/2) + log((8*a*b^
5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3 - ((((((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3
*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2)*(32*a^4*b^3 - 32*b^7 + 32*a*b^2*d*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)
^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) + (16*(a +
b*cot(c + d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/d^2)*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^
2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2))/2)*((20*a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b
^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2)/(4*d^4) + (5*a*b^6)/(4*d^2) - (5*a^3*b^4)/(2*d^2) + (a^5*b^2
)/(4*d^2))^(1/2) - log(((((-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^
5*b^2*d^2)/d^4)^(1/2)*(32*b^7 - 32*a^4*b^3 + 32*a*b^2*d*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a
*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) + (16*(a + b*cot(c + d*
x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/d^2)*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) -
5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2))/2 + (8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*(-((20*
a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^
4*d^2 - a^5*b^2*d^2)/(4*d^4))^(1/2) + log((8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3 - ((((-((-b^6*d^4*(5*a^4 +
b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(32*a^4*b^3 - 32*b^7 + 32
*a*b^2*d*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(
1/2)*(a + b*cot(c + d*x))^(1/2)))/(2*d) + (16*(a + b*cot(c + d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6
*b^4))/d^2)*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4
)^(1/2))/2)*((5*a*b^6)/(4*d^2) - (20*a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6
*d^4)^(1/2)/(4*d^4) - (5*a^3*b^4)/(2*d^2) + (a^5*b^2)/(4*d^2))^(1/2) - log(((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*
a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a
^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 + 32*a*b^2*d*
(((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(a
+ b*cot(c + d*x))^(1/2)))/(2*d) + (16*a^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)
)/d^2))/2 - (8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3)*(-(a^7*d^2 - (20*a^6*b^8*d^4 - a^4*b^10*d^4 - 110*a^8
*b^6*d^4 + 100*a^10*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2) + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/(4*d^4))^(1/2) - (2*b*(
a + b*cot(c + d*x))^(5/2))/(5*d) + (4*a^2*b*(a + b*cot(c + d*x))^(1/2))/d