3.2.3 \(\int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx\) [103]

Optimal. Leaf size=185 \[ \frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}} \]

[Out]

(I*A+B)*arctanh((a+b*cot(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-(I*A-B)*arctanh((a+b*cot(d*x+c))^(1/2)/(
a+I*b)^(1/2))/(a+I*b)^(5/2)/d+2/3*(A*b-B*a)/(a^2+b^2)/d/(a+b*cot(d*x+c))^(3/2)+2*(2*A*a*b-B*a^2+B*b^2)/(a^2+b^
2)^2/d/(a+b*cot(d*x+c))^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3610, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (A b-a B)}{3 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (a^2 (-B)+2 a A b+b^2 B\right )}{d \left (a^2+b^2\right )^2 \sqrt {a+b \cot (c+d x)}}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {(-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a +
b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*(A*b - a*B))/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3
/2)) + (2*(2*a*A*b - a^2*B + b^2*B))/((a^2 + b^2)^2*d*Sqrt[a + b*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 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 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 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*} \int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx &=\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {\int \frac {a A+b B-(A b-a B) \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx}{a^2+b^2}\\ &=\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}}+\frac {\int \frac {a^2 A-A b^2+2 a b B-\left (2 a A b-a^2 B+b^2 B\right ) \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}}+\frac {(A-i B) \int \frac {1+i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx}{2 (a-i b)^2}+\frac {(A+i B) \int \frac {1-i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}} \, dx}{2 (a+i b)^2}\\ &=\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}}+\frac {(i A-B) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \cot (c+d x)\right )}{2 (a+i b)^2 d}-\frac {(i A+B) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \cot (c+d x)\right )}{2 (a-i b)^2 d}\\ &=\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}}-\frac {(A-i B) \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 (i a+b)^2 d}-\frac {(A+i B) \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 )}{(i a-b)^2 b d}\\ &=\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {2 (A b-a B)}{3 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^{3/2}}+\frac {2 \left (2 a A b-a^2 B+b^2 B\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \cot (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 3.66, size = 319, normalized size = 1.72 \begin {gather*} -\frac {\frac {3 \left (2 a b \left (A \sqrt {-b^2}+b B\right )+a^2 \left (A b-\sqrt {-b^2} B\right )+b^2 \left (-A b+\sqrt {-b^2} B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}+\frac {3 \left (2 a b \left (A \sqrt {-b^2}-b B\right )-a^2 \left (A b+\sqrt {-b^2} B\right )+b^2 \left (A b+\sqrt {-b^2} B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}+\frac {2 \left (a^2+b^2\right ) (-A b+a B)}{(a+b \cot (c+d x))^{3/2}}+\frac {6 \left (-2 a A b+a^2 B-b^2 B\right )}{\sqrt {a+b \cot (c+d x)}}}{3 \left (a^2+b^2\right )^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((3*(2*a*b*(A*Sqrt[-b^2] + b*B) + a^2*(A*b - Sqrt[-b^2]*B) + b^2*(-(A*b) + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a
+ b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) + (3*(2*a*b*(A*Sqrt[-b^2] - b*B) -
a^2*(A*b + Sqrt[-b^2]*B) + b^2*(A*b + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(
Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*(a^2 + b^2)*(-(A*b) + a*B))/(a + b*Cot[c + d*x])^(3/2) + (6*(-2*a*A*b +
a^2*B - b^2*B))/Sqrt[a + b*Cot[c + d*x]])/((a^2 + b^2)^2*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3235\) vs. \(2(161)=322\).
time = 0.66, size = 3236, normalized size = 17.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3236\)
default	\(\text {Expression too large to display}\)	\(3236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/3*(A*b-B*a)/(a^2+b^2)/(a+b*cot(d*x+c))^(3/2)+2*(2*A*a*b-B*a^2+B*b^2)/(a^2+b^2)^2/(a+b*cot(d*x+c))^(1/2)
-2/(a^2+b^2)^2*(1/4/b/(5*a^4-10*a^2*b^2+b^4)/(a^2+b^2)^(3/2)*(1/2*(3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)
^(3/2)*a^6+5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^4*b^2+A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)
^(3/2)*a^2*b^4-A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*b^6+2*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)
^(1/2)*a^8-18*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^6*b^2-10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+
b^2)^(1/2)*a^4*b^4+10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2*b^6-5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2
)*a^9+20*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^7*b^2-6*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5*b^4-28*A*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)*a^3*b^6+3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^8+10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(
1/2)*a^7*b-10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^5*b^3-18*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+
b^2)^(1/2)*a^3*b^5+2*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^7-15*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*
a^8*b+20*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^6*b^3+22*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4*b^5-12*B*(2*(a^2+b^2)^
(1/2)+2*a)^(1/2)*a^2*b^7+B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^9)*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*(30*A*a^8*b^2-40*A*a^6*b^4-44*A*a^4*b^6+24*A*a^2*b^8-2*A*b^10-10*B*a
^9*b+40*B*a^7*b^3-12*B*a^5*b^5-56*B*a^3*b^7+6*B*a*b^9-1/2*(3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a
^6+5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^4*b^2+A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a
^2*b^4-A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*b^6+2*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a
^8-18*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^6*b^2-10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/
2)*a^4*b^4+10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2*b^6-5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^9+20
*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^7*b^2-6*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5*b^4-28*A*(2*(a^2+b^2)^(1/2)+2*a
)^(1/2)*a^3*b^6+3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^8+10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^7
*b-10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^5*b^3-18*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/
2)*a^3*b^5+2*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^7-15*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^8*b+20
*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^6*b^3+22*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4*b^5-12*B*(2*(a^2+b^2)^(1/2)+2*
a)^(1/2)*a^2*b^7+B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^9)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(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)))+1/4/b/(5*
a^4-10*a^2*b^2+b^4)/(a^2+b^2)^(3/2)*(-1/2*(3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^6+5*A*(2*(a^2+b
^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^4*b^2+A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^2*b^4-A*(2*(a^2
+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*b^6+2*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^8-18*A*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^6*b^2-10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4*b^4+10*A*
(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2*b^6-5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^9+20*A*(2*(a^2+b^2)^
(1/2)+2*a)^(1/2)*a^7*b^2-6*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5*b^4-28*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3*b^6+
3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^8+10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^7*b-10*B*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^5*b^3-18*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3*b^5+2*B*(
2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^7-15*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^8*b+20*B*(2*(a^2+b^2)^
(1/2)+2*a)^(1/2)*a^6*b^3+22*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4*b^5-12*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2*b^7
+B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^9)*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-(
a^2+b^2)^(1/2))+2*(-30*A*a^8*b^2+40*A*a^6*b^4+44*A*a^4*b^6-24*A*a^2*b^8+2*A*b^10+10*B*a^9*b-40*B*a^7*b^3+12*B*
a^5*b^5+56*B*a^3*b^7-6*B*a*b^9+1/2*(3*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^6+5*A*(2*(a^2+b^2)^(1/
2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^4*b^2+A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*a^2*b^4-A*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)*(a^2+b^2)^(3/2)*b^6+2*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^8-18*A*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^6*b^2-10*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4*b^4+10*A*(2*(a^2
+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2*b^6-5*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^9+20*A*(2*(a^2+b^2)^(1/2)+2
*a)^(1/2)*a^7*b^2-6*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5*b^4-28*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3*b^6+3*A*(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^8+10*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^7*b-10*B*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^5*b^3-18*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3*b^5+2*B*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^7-15*...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/(b*cot(d*x + c) + a)^(5/2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \cot {\left (c + d x \right )}}{\left (a + b \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/(b*cot(d*x + c) + a)^(5/2), x)

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Mupad [B]
time = 17.93, size = 2500, normalized size = 13.51 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((((a + b*cot(c + d*x))^(1/2)*(320*A^2*a^4*b^14*d^3 - 16*A^2*b^18*d^3 + 1024*A^2*a^6*b^12*d^3 + 1440*A^2*a
^8*b^10*d^3 + 1024*A^2*a^10*b^8*d^3 + 320*A^2*a^12*b^6*d^3 - 16*A^2*a^16*b^2*d^3) + ((((320*A^4*a^2*b^8*d^4 -
16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 4
0*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 +
5*a^8*b^2*d^4))^(1/2)*(896*A*a^6*b^15*d^4 - ((((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4
+ 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(
a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*cot(c + d
*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16
128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b
^2*d^5))/4 - 160*A*a^2*b^19*d^4 - 128*A*a^4*b^17*d^4 - 32*A*b^21*d^4 + 3136*A*a^8*b^13*d^4 + 4928*A*a^10*b^11*
d^4 + 4480*A*a^12*b^9*d^4 + 2432*A*a^14*b^7*d^4 + 736*A*a^16*b^5*d^4 + 96*A*a^18*b^3*d^4))/4)*(((320*A^4*a^2*b
^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^
5*d^2 + 40*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*
b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 - 96*A^3*a^3*b^13*d^2 - 240*A^3*a^5*b^11*d^2 - 320*A^3*a^7*b^9*d^2 - 240*A^
3*a^9*b^7*d^2 - 96*A^3*a^11*b^5*d^2 - 16*A^3*a^13*b^3*d^2 - 16*A^3*a*b^15*d^2)*(((320*A^4*a^2*b^8*d^4 - 16*A^4
*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A^2*
a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8
*b^2*d^4))^(1/2))/4 + (log((((a + b*cot(c + d*x))^(1/2)*(320*A^2*a^4*b^14*d^3 - 16*A^2*b^18*d^3 + 1024*A^2*a^6
*b^12*d^3 + 1440*A^2*a^8*b^10*d^3 + 1024*A^2*a^10*b^8*d^3 + 320*A^2*a^12*b^6*d^3 - 16*A^2*a^16*b^2*d^3) + ((-(
(320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1
/2) + 4*A^2*a^5*d^2 - 40*A^2*a^3*b^2*d^2 + 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6
*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(896*A*a^6*b^15*d^4 - ((-((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4
- 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) + 4*A^2*a^5*d^2 - 40*A^2*a^3*b^2*d
^2 + 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4)
)^(1/2)*(a + b*cot(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 +
13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*
a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 - 160*A*a^2*b^19*d^4 - 128*A*a^4*b^17*d^4 - 32*A*b^21*d^4 + 3136*A*a^8*b^13
*d^4 + 4928*A*a^10*b^11*d^4 + 4480*A*a^12*b^9*d^4 + 2432*A*a^14*b^7*d^4 + 736*A*a^16*b^5*d^4 + 96*A*a^18*b^3*d
^4))/4)*(-((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*
b^2*d^4)^(1/2) + 4*A^2*a^5*d^2 - 40*A^2*a^3*b^2*d^2 + 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 +
10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 - 96*A^3*a^3*b^13*d^2 - 240*A^3*a^5*b^11*d^2 - 320
*A^3*a^7*b^9*d^2 - 240*A^3*a^9*b^7*d^2 - 96*A^3*a^11*b^5*d^2 - 16*A^3*a^13*b^3*d^2 - 16*A^3*a*b^15*d^2)*(-((32
0*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2)
+ 4*A^2*a^5*d^2 - 40*A^2*a^3*b^2*d^2 + 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^
4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 - log(- ((a + b*cot(c + d*x))^(1/2)*(320*A^2*a^4*b^14*d^3 - 16*A
^2*b^18*d^3 + 1024*A^2*a^6*b^12*d^3 + 1440*A^2*a^8*b^10*d^3 + 1024*A^2*a^10*b^8*d^3 + 320*A^2*a^12*b^6*d^3 - 1
6*A^2*a^16*b^2*d^3) - (((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 -
400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(16*a^10*d^4 + 16*b^10*d^
4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*((((320*A^4*a^2*b^8*d^4 - 16*A
^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A^
2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*
d^4 + 80*a^8*b^2*d^4))^(1/2)*(a + b*cot(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5
+ 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 288
0*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5) - 32*A*b^21*d^4 - 160*A*a^2*b^19*d^4 - 128*A*a^4*b^17*d^4
+ 896*A*a^6*b^15*d^4 + 3136*A*a^8*b^13*d^4 + 4928*A*a^10*b^11*d^4 + 4480*A*a^12*b^9*d^4 + 2432*A*a^14*b^7*d^4
+ 736*A*a^16*b^5*d^4 + 96*A*a^18*b^3*d^4))*(((...

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