3.105 \(\int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=132 \[ -\frac{4 a b}{d \left (a^2+b^2\right ) \sqrt{a+b \cot (c+d x)}}-\frac{(-b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{(b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]
[Out]
-(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a
+ b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])
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Rubi [A] time = 0.248112, antiderivative size = 132, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used =
{3529, 3539, 3537, 63, 208} \[ -\frac{4 a b}{d \left (a^2+b^2\right ) \sqrt{a+b \cot (c+d x)}}-\frac{(-b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{(b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]
[Out]
-(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a
+ b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])
Rule 3529
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 3539
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]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx &=-\frac{4 a b}{\left (a^2+b^2\right ) d \sqrt{a+b \cot (c+d x)}}+\frac{\int \frac{-a^2+b^2+2 a b \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx}{a^2+b^2}\\ &=-\frac{4 a b}{\left (a^2+b^2\right ) 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)}-\frac{(a+i b) \int \frac{1+i \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx}{2 (a-i b)}\\ &=-\frac{4 a b}{\left (a^2+b^2\right ) d \sqrt{a+b \cot (c+d x)}}-\frac{(a+i b) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \cot (c+d x)\right )}{2 (i a+b) d}-\frac{(i a+b) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \cot (c+d x)\right )}{2 (a+i b) d}\\ &=-\frac{4 a b}{\left (a^2+b^2\right ) d \sqrt{a+b \cot (c+d x)}}-\frac{(a-i b) \operatorname{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 )}{(a+i b) b d}-\frac{(a+i b) \operatorname{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 )}{(a-i b) 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)^{3/2} d}+\frac{(i a+b) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{3/2} d}-\frac{4 a b}{\left (a^2+b^2\right ) d \sqrt{a+b \cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.40507, size = 216, normalized size = 1.64 \[ \frac{\sin (c+d x) (a-b \cot (c+d x)) \left (\sqrt{a-i b} \left (i (a-i b)^2 \sqrt{a+b \cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right )-4 a b \sqrt{a+i b}\right )-i (a+i b)^{5/2} \sqrt{a+b \cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right )\right )}{d (a-i b)^{3/2} (a+i b)^{3/2} \sqrt{a+b \cot (c+d x)} (a \sin (c+d x)-b \cos (c+d x))} \]
Antiderivative was successfully verified.
[In]
Integrate[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]
[Out]
((a - b*Cot[c + d*x])*((-I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]]*Sqrt[a + b*Cot[c +
d*x]] + Sqrt[a - I*b]*(-4*a*Sqrt[a + I*b]*b + I*(a - I*b)^2*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]]*S
qrt[a + b*Cot[c + d*x]]))*Sin[c + d*x])/((a - I*b)^(3/2)*(a + I*b)^(3/2)*d*Sqrt[a + b*Cot[c + d*x]]*(-(b*Cos[c
+ d*x]) + a*Sin[c + d*x]))
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Maple [B] time = 0.056, size = 2291, normalized size = 17.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-a+b*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x)
[Out]
-4*a*b/(a^2+b^2)/d/(a+b*cot(d*x+c))^(1/2)-1/4/d*b^3/(a^2+b^2)^2*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^3/(a^2+b^2)^(3/2)/(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))
+2/d*b^5/(a^2+b^2)^(5/2)/(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))+1/d*b^3/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*c
ot(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))-2/d*b^5/(a^2+b^2)^(5/2)/(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/d*b^3/(a^2+b^2)^2*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*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/d/b/(a^2+b^2)^(5/2)/(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))*a^6-1/d*b^3/(a^2+b^2)^(5/2)/(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))*a^2-3/4/d*b^3/(a^2+b^2)^(5/2)*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*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+2/d*b^3/(a^2+b^2)^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))*a+1/4/d/b/(a^2+b^2)^2*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^4-2/d*b^3/(a^2+b^2)^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))*a+2/d*b/(a^2+b^2)^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))*a^3-2/d*b/(a^2+b^2)^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))*a^3+4/d*b/(a^2+b^2)^(5/2)/(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))*a^4+1/2/d*b/(a^2+b^2)^(5/2)*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^3-1/2/d*b/(a^2+b^2)^(5/2)*l
n((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*(a^2+b^2)^(1/2)+2*a)
^(1/2)*a^3-4/d*b/(a^2+b^2)^(5/2)/(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*co
t(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4+1/d/b/(a^2+b^2)^(3/2)/(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))*a^4+1/d/b/(a^2+b^2)^(
5/2)/(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))*a^6+1/d*b^3/(a^2+b^2)^(5/2)/(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))*a^2-1/4/d/b/(a^2+b^2)^(5/2)*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^5+3/4/d*b
^3/(a^2+b^2)^(5/2)*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-1/d/b/(a^2+b^2)^(3/2)/(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))*a^4-1/4/d/b/(a^2+b^2)^2*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*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+1/4/d/b/
(a^2+b^2)^(5/2)*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*(a^
2+b^2)^(1/2)+2*a)^(1/2)*a^5
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \cot \left (d x + c\right ) - a}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-a+b*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-a+b*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a}{a \sqrt{a + b \cot{\left (c + d x \right )}} + b \sqrt{a + b \cot{\left (c + d x \right )}} \cot{\left (c + d x \right )}}\, dx - \int - \frac{b \cot{\left (c + d x \right )}}{a \sqrt{a + b \cot{\left (c + d x \right )}} + b \sqrt{a + b \cot{\left (c + d x \right )}} \cot{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-a+b*cot(d*x+c))/(a+b*cot(d*x+c))**(3/2),x)
[Out]
-Integral(a/(a*sqrt(a + b*cot(c + d*x)) + b*sqrt(a + b*cot(c + d*x))*cot(c + d*x)), x) - Integral(-b*cot(c + d
*x)/(a*sqrt(a + b*cot(c + d*x)) + b*sqrt(a + b*cot(c + d*x))*cot(c + d*x)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \cot \left (d x + c\right ) - a}{{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-a+b*cot(d*x+c))/(a+b*cot(d*x+c))^(3/2),x, algorithm="giac")
[Out]
integrate((b*cot(d*x + c) - a)/(b*cot(d*x + c) + a)^(3/2), x)