3.1.11 \(\int (c \cot (a+b x))^{3/2} \, dx\) [11]

Optimal. Leaf size=210 \[ -\frac {c^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b} \]

[Out]

-1/2*c^(3/2)*arctan(1-2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))/b*2^(1/2)+1/2*c^(3/2)*arctan(1+2^(1/2)*(c*cot(b*x+
a))^(1/2)/c^(1/2))/b*2^(1/2)-1/4*c^(3/2)*ln(c^(1/2)+cot(b*x+a)*c^(1/2)-2^(1/2)*(c*cot(b*x+a))^(1/2))/b*2^(1/2)
+1/4*c^(3/2)*ln(c^(1/2)+cot(b*x+a)*c^(1/2)+2^(1/2)*(c*cot(b*x+a))^(1/2))/b*2^(1/2)-2*c*(c*cot(b*x+a))^(1/2)/b

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Rubi [A]
time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {c^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}+1\right )}{\sqrt {2} b}-\frac {c^{3/2} \log \left (\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Cot[a + b*x])^(3/2),x]

[Out]

-((c^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b)) + (c^(3/2)*ArcTan[1 + (Sqrt[2]*Sqr
t[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (2*c*Sqrt[c*Cot[a + b*x]])/b - (c^(3/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a
+ b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b) + (c^(3/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]
*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int (c \cot (a+b x))^{3/2} \, dx &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx\\ &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {c^3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b}\\ &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {c^2 \text {Subst}\left (\int \frac {c-x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b}+\frac {c^2 \text {Subst}\left (\int \frac {c+x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b}\\ &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}+2 x}{-c-\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}-2 x}{-c+\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{c-\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{c+\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b}\\ &=-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}\\ &=-\frac {c^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 159, normalized size = 0.76 \begin {gather*} -\frac {(c \cot (a+b x))^{3/2} \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (a+b x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (a+b x)}\right )+8 \sqrt {\cot (a+b x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )\right )}{4 b \cot ^{\frac {3}{2}}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Cot[a + b*x])^(3/2),x]

[Out]

-1/4*((c*Cot[a + b*x])^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*
Sqrt[Cot[a + b*x]]] + 8*Sqrt[Cot[a + b*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]] - Sqrt
[2]*Log[1 + Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]))/(b*Cot[a + b*x]^(3/2))

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Maple [A]
time = 0.27, size = 149, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\)	\(149\)
default	\(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b*c*((c*cot(b*x+a))^(1/2)-1/8*(c^2)^(1/4)*2^(1/2)*(ln((c*cot(b*x+a)+(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)*2^(1/2
)+(c^2)^(1/2))/(c*cot(b*x+a)-(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)*2^(1/2)+(c^2)^(1/2)))+2*arctan(2^(1/2)/(c^2)^(1/
4)*(c*cot(b*x+a))^(1/2)+1)-2*arctan(-2^(1/2)/(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)+1)))

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Maxima [A]
time = 0.50, size = 179, normalized size = 0.85 \begin {gather*} \frac {{\left (2 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + 2 \, \sqrt {2} \sqrt {c} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + \sqrt {2} \sqrt {c} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - \sqrt {2} \sqrt {c} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - 8 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )} c}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cot(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*sqrt(c)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(c) + 2*sqrt(c/tan(b*x + a)))/sqrt(c)) + 2*sqrt(2)*sqrt
(c)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(c) - 2*sqrt(c/tan(b*x + a)))/sqrt(c)) + sqrt(2)*sqrt(c)*log(sqrt(2)*sqrt
(c)*sqrt(c/tan(b*x + a)) + c + c/tan(b*x + a)) - sqrt(2)*sqrt(c)*log(-sqrt(2)*sqrt(c)*sqrt(c/tan(b*x + a)) + c
+ c/tan(b*x + a)) - 8*sqrt(c/tan(b*x + a)))*c/b

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   catdef: division by zero

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*cot(b*x + a))^(3/2), x)

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Mupad [B]
time = 0.37, size = 75, normalized size = 0.36 \begin {gather*} -\frac {2\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cot(a + b*x))^(3/2),x)

[Out]

- (2*c*(c*cot(a + b*x))^(1/2))/b - ((-1)^(1/4)*c^(3/2)*atan(((-1)^(1/4)*(c*cot(a + b*x))^(1/2))/c^(1/2))*1i)/b
- ((-1)^(1/4)*c^(3/2)*atanh(((-1)^(1/4)*(c*cot(a + b*x))^(1/2))/c^(1/2))*1i)/b

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