3.1.96 \(\int (a+b \cot (c+d x))^{3/2} (A+B \cot (c+d x)) \, dx\) [96]

Optimal. Leaf size=150 \[ \frac {(a-i b)^{3/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 (A b+a B) \sqrt {a+b \cot (c+d x)}}{d}-\frac {2 B (a+b \cot (c+d x))^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 150, 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 = {3609, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 (a B+A b) \sqrt {a+b \cot (c+d x)}}{d}+\frac {(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 B (a+b \cot (c+d x))^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x])^(3/2)*(A + B*Cot[c + d*x]),x]

[Out]

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

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Mathematica [A]
time = 1.01, size = 294, normalized size = 1.96 \begin {gather*} -\frac {\frac {3 \sqrt {a-\sqrt {-b^2}} \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 )}{b^2+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}}}+6 (A b+a B) \sqrt {a+b \cot (c+d x)}+2 B (a+b \cot (c+d x))^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[c + d*x])^(3/2)*(A + B*Cot[c + d*x]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(919\) vs. \(2(126)=252\).
time = 0.66, size = 920, normalized size = 6.13

method	result	size
derivativedivides	\(\frac {-\frac {2 B \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 A b \sqrt {a +b \cot \left (d x +c \right )}-2 B a \sqrt {a +b \cot \left (d x +c \right )}-\frac {\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 A \sqrt {a^{2}+b^{2}}\, b^{2}-2 B \sqrt {a^{2}+b^{2}}\, a b -\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 b}-\frac {-\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 A \sqrt {a^{2}+b^{2}}\, b^{2}+2 B \sqrt {a^{2}+b^{2}}\, a b +\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \cot \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 b}}{d}\)	\(920\)
default	\(\frac {-\frac {2 B \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 A b \sqrt {a +b \cot \left (d x +c \right )}-2 B a \sqrt {a +b \cot \left (d x +c \right )}-\frac {\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 A \sqrt {a^{2}+b^{2}}\, b^{2}-2 B \sqrt {a^{2}+b^{2}}\, a b -\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 b}-\frac {-\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 A \sqrt {a^{2}+b^{2}}\, b^{2}+2 B \sqrt {a^{2}+b^{2}}\, a b +\frac {\left (-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a +A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}-A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}+B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b -2 B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \cot \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 b}}{d}\)	\(920\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))^(3/2)*(A+B*cot(d*x+c)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B \cot {\left (c + d x \right )}\right ) \left (a + b \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c))**(3/2)*(A+B*cot(d*x+c)),x)

[Out]

Integral((A + B*cot(c + d*x))*(a + b*cot(c + d*x))**(3/2), 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))^(3/2)*(A+B*cot(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*cot(d*x + c) + A)*(b*cot(d*x + c) + a)^(3/2), x)

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Mupad [B]
time = 13.76, size = 2823, normalized size = 18.82 \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))^(3/2),x)

[Out]

log((16*A^3*a*b^3*(a^2 + b^2)^2)/d^3 - (((16*b^2*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*
a*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^2*b + a*d*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^
2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d + (16*A^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b
^2))/d^2)*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2)*((6*A^4*a^2*b
^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a^3)/(4*d^2) + (3*A^2*a*b^2)/(4*d^2))^(1/2) - l
og((16*A^3*a*b^3*(a^2 + b^2)^2)/d^3 - (((16*b^2*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a
*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^2*b - a*d*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2
*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d - (16*A^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^
2))/d^2)*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2)*(((6*A^4*a^2*b
^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/(4*d^4))^(1/2) - log((16*A^3*
a*b^3*(a^2 + b^2)^2)/d^3 - (((16*b^2*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/
d^4)^(1/2)*(A*b^3 + A*a^2*b - a*d*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4
)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d - (16*A^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2)*
(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2)*(-((6*A^4*a^2*b^4*d^4
- A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/(4*d^4))^(1/2) + log((16*A^3*a*b^3*(
a^2 + b^2)^2)/d^3 - (((16*b^2*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1
/2)*(A*b^3 + A*a^2*b + a*d*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2)
*(a + b*cot(c + d*x))^(1/2)))/d + (16*A^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2)*(-((-A^
4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2)*((3*A^2*a*b^2)/(4*d^2) - (A^2
*a^3)/(4*d^2) - (6*A^4*a^2*b^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2)/(4*d^4))^(1/2) - log((8*B^3*b^2*(a
^2 - b^2)*(a^2 + b^2)^2)/d^3 - (((16*B^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 + (16*a*b
^2*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 - d*(((-B
^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*
(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2))/2)*(((6*B^4*a^2*b^4*d^4 -
B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/(4*d^4))^(1/2) - log((8*B^3*b^2*(a^2 -
b^2)*(a^2 + b^2)^2)/d^3 - (((16*B^2*b^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 + (16*a*b^2*(
-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 - d*(-((-B^4
*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*(-
((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2))/2)*(-((6*B^4*a^2*b^4*d^4 -
B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/(4*d^4))^(1/2) + log((((16*B^2*b^2*(a
+ b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 - (16*a*b^2*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*
a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 + d*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2
- 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3
*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2))/2 + (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*((6*B^4*a^2*b^4*d^4 - B^4*b
^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a^3)/(4*d^2) - (3*B^2*a*b^2)/(4*d^2))^(1/2) + log((((16*B^2*b
^2*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 - (16*a*b^2*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2)
- B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 + d*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*
a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2)
- B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2))/2 + (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*((B^2*a^3)/(4*d^2)
- (6*B^4*a^2*b^4*d^4 - B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2)/(4*d^4) - (3*B^2*a*b^2)/(4*d^2))^(1/2) - (2*B*(
a + b*cot(c + d*x))^(3/2))/(3*d) - (2*A*b*(a + b*cot(c + d*x))^(1/2))/d - (2*B*a*(a + b*cot(c + d*x))^(1/2))/d

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