3.35 \(\int \frac{1}{(a+b \cot ^2(c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{b \cot (c+d x)}{a d (a-b) \sqrt{a+b \cot ^2(c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{3/2}} \]

[Out]

-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a -
b)*d*Sqrt[a + b*Cot[c + d*x]^2])

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Rubi [A]  time = 0.0612232, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3661, 382, 377, 203} \[ \frac{b \cot (c+d x)}{a d (a-b) \sqrt{a+b \cot ^2(c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x]^2)^(-3/2),x]

[Out]

-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a -
b)*d*Sqrt[a + b*Cot[c + d*x]^2])

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

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

Rubi steps

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

Mathematica [C]  time = 3.66013, size = 231, normalized size = 2.72 \[ -\frac{\cos ^2(c+d x) \cot (c+d x) \left (4 (a-b)^2 \cos ^2(c+d x) \left (a \tan ^2(c+d x)+b\right ) \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \cos ^2(c+d x)}{a}\right )-\frac{15 a \left (3 a \tan ^2(c+d x)+2 b\right ) \left (\left (a \tan ^2(c+d x)+b\right ) \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(c+d x)}{a}}\right )-a \sec ^2(c+d x) \sqrt{\frac{(a-b) \cos ^4(c+d x) \left (a \tan ^2(c+d x)+b\right )}{a^2}}\right )}{\sqrt{\frac{(a-b) \cos ^4(c+d x) \left (a \tan ^2(c+d x)+b\right )}{a^2}}}\right )}{15 a^3 d (a-b) \sqrt{a+b \cot ^2(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Cos[c + d*x]^2*Cot[c + d*x]*(4*(a - b)^2*Cos[c + d*x]^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[c + d*x]^2
)/a]*(b + a*Tan[c + d*x]^2) - (15*a*(2*b + 3*a*Tan[c + d*x]^2)*(ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*(b +
a*Tan[c + d*x]^2) - a*Sec[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]))/Sqrt[((a - b)
*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]))/(15*a^3*(a - b)*d*Sqrt[a + b*Cot[c + d*x]^2])

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Maple [A]  time = 0.028, size = 104, normalized size = 1.2 \begin{align*} -{\frac{1}{d \left ( a-b \right ) ^{2}{b}^{2}}\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( dx+c \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \right ) }+{\frac{b\cot \left ( dx+c \right ) }{a \left ( a-b \right ) d}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cot(d*x+c)^2)^(3/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.30348, size = 1169, normalized size = 13.75 \begin{align*} \left [-\frac{{\left (a^{2} + a b -{\left (a^{2} - a b\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt{-a + b} \log \left (-2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \,{\left ({\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - b\right )} \sqrt{-a + b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right ) + a^{2} - 2 \, b^{2} + 4 \,{\left (a b - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right ) + 4 \,{\left (a b - b^{2}\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{4 \,{\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) -{\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}, \frac{{\left (a^{2} + a b -{\left (a^{2} - a b\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt{a - b} \arctan \left (-\frac{\sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - b}\right ) - 2 \,{\left (a b - b^{2}\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{2 \,{\left ({\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} d \cos \left (2 \, d x + 2 \, c\right ) -{\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right )} d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*((a^2 + a*b - (a^2 - a*b)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 +
2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))
*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*d*x + 2*c)) + 4*(a*b - b^2)*sqrt(((a - b)*cos(2*d*x + 2*
c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c)
- (a^4 - a^3*b - a^2*b^2 + a*b^3)*d), 1/2*((a^2 + a*b - (a^2 - a*b)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt
(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x +
2*c) - b)) - 2*(a*b - b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/(
(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cot(d*x+c)**2)**(3/2),x)

[Out]

Integral((a + b*cot(c + d*x)**2)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(3/2),x, algorithm="giac")

[Out]

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