3.33 \(\int \sqrt {a+b \cot ^2(c+d x)} \, dx\)

Optimal. Leaf size=87 \[ -\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3661, 402, 217, 206, 377, 203} \[ -\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cot[c + d*x]^2],x]

[Out]

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

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])

Rule 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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])

Rubi steps

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

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Mathematica [C]  time = 0.61, size = 202, normalized size = 2.32 \[ \frac {i \left (\sqrt {a-b} \log \left (-\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \cot ^2(c+d x)}+a-i b \cot (c+d x)\right )}{(a-b)^{3/2} (\cot (c+d x)+i)}\right )-\sqrt {a-b} \log \left (\frac {4 i \left (\sqrt {a-b} \sqrt {a+b \cot ^2(c+d x)}+a+i b \cot (c+d x)\right )}{(a-b)^{3/2} (\cot (c+d x)-i)}\right )+2 i \sqrt {b} \log \left (\sqrt {b} \sqrt {a+b \cot ^2(c+d x)}+b \cot (c+d x)\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Cot[c + d*x]^2],x]

[Out]

((I/2)*(Sqrt[a - b]*Log[((-4*I)*(a - I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(3/2
)*(I + Cot[c + d*x]))] - Sqrt[a - b]*Log[((4*I)*(a + I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]
))/((a - b)^(3/2)*(-I + Cot[c + d*x]))] + (2*I)*Sqrt[b]*Log[b*Cot[c + d*x] + Sqrt[b]*Sqrt[a + b*Cot[c + d*x]^2
]]))/d

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fricas [B]  time = 0.94, size = 703, normalized size = 8.08 \[ \left [\frac {\sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\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 ) + b\right ) + \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {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 \, b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}\right )}{2 \, d}, -\frac {2 \, \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 ) + a - b}\right ) - \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {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 \, b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}\right )}{2 \, d}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-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 )}{b \cos \left (2 \, d x + 2 \, c\right ) + b}\right ) + \sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\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 ) + b\right )}{2 \, d}, -\frac {\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 ) + a - b}\right ) - \sqrt {-b} \arctan \left (\frac {\sqrt {-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 )}{b \cos \left (2 \, d x + 2 \, c\right ) + b}\right )}{d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + 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) + b) + sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*co
s(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1)))/d, -1/2*(
2*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) + a - b)) - sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*c
os(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1)))/d, 1/2*(
2*sqrt(-b)*arctan(sqrt(-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)/(b
*cos(2*d*x + 2*c) + b)) + sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + 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) + b))/d, -(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) + a - b)) - sqr
t(-b)*arctan(sqrt(-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)/(b*cos(
2*d*x + 2*c) + b)))/d]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(sin(d*x+c))]Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2
)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_no
step/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*p
i/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign
: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Warning, replacing 0 by ` u`, a substitution variable should perhaps be
purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by
` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable s
hould perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning
, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substi
tution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps
be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, integration
of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warni
ng, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular valu
e [0] was discarded and replaced randomly by 0=[-92]Warning, need to choose a branch for the root of a polynom
ial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[4]War
ning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular va
lue [0] was discarded and replaced randomly by 0=[69]Warning, need to choose a branch for the root of a polyno
mial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[-99]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular
value [0] was discarded and replaced randomly by 0=[94]Warning, need to choose a branch for the root of a pol
ynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[-
94]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regu
lar value [0] was discarded and replaced randomly by 0=[-41]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by
0=[-81]Precision problem choosing root in common_EXT, current precision 14Evaluation time: 0.63index.cc index_
m operator + Error: Bad Argument Value

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maple [B]  time = 0.42, size = 170, normalized size = 1.95 \[ -\frac {\sqrt {b}\, \ln \left (\cot \left (d x +c \right ) \sqrt {b}+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{d}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{d b \left (a -b \right )}-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{d \,b^{2} \left (a -b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is b-a positive or negative?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cot(c + d*x)^2)^(1/2),x)

[Out]

int((a + b*cot(c + d*x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cot ^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*cot(c + d*x)**2), x)

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