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

Optimal. Leaf size=81 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0496097, antiderivative size = 81, normalized size of antiderivative = 1., 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 = {4128, 402, 217, 206, 377, 203} \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Csc[c + d*x]^2],x]

[Out]

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

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

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

Mathematica [A]  time = 0.233168, size = 149, normalized size = 1.84 \[ \frac{\sqrt{2} \sin (c+d x) \sqrt{a+b \csc ^2(c+d x)} \left (\sqrt{a} \log \left (\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right )-\sqrt{-b} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{-b} \cos (c+d x)}{\sqrt{a \cos (2 (c+d x))-a-2 b}}\right )\right )}{d \sqrt{a \cos (2 (c+d x))-a-2 b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*Sqrt[a + b*Csc[c + d*x]^2]*(-(Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[c + d*x])/Sqrt[-a - 2*b + a*Cos[
2*(c + d*x)]]]) + Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])*Sin[c + d*x
])/(d*Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]])

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Maple [B]  time = 0.398, size = 419, normalized size = 5.2 \begin{align*} -{\frac{\sqrt{4} \left ( -1+\cos \left ( dx+c \right ) \right ) }{4\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1}}} \left ( \ln \left ( -2\,{\frac{-1+\cos \left ( dx+c \right ) }{\sqrt{b} \left ( \sin \left ( dx+c \right ) \right ) ^{2}} \left ( \cos \left ( dx+c \right ) \sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}\sqrt{b}+a\cos \left ( dx+c \right ) +\sqrt{b}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}+a+b \right ) } \right ) \sqrt{b}\sqrt{-a}-\sqrt{b}\ln \left ( -4\,{\frac{1}{-1+\cos \left ( dx+c \right ) } \left ( \cos \left ( dx+c \right ) \sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}\sqrt{b}-a\cos \left ( dx+c \right ) +\sqrt{b}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}+a+b \right ) } \right ) \sqrt{-a}-2\,a\ln \left ( 4\,\cos \left ( dx+c \right ) \sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}-4\,a\cos \left ( dx+c \right ) +4\,\sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}} \right ) \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d*4^(1/2)/(-a)^(1/2)*((a*cos(d*x+c)^2-a-b)/(cos(d*x+c)^2-1))^(1/2)*(-1+cos(d*x+c))*(ln(-2/b^(1/2)*(-1+cos
(d*x+c))*(cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*b^(1/2)+a*cos(d*x+c)+b^(1/2)*(-(a*cos(d*x+
c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)+a+b)/sin(d*x+c)^2)*b^(1/2)*(-a)^(1/2)-b^(1/2)*ln(-4*(cos(d*x+c)*(-(a*cos(d*x
+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*b^(1/2)-a*cos(d*x+c)+b^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)
+a+b)/(-1+cos(d*x+c)))*(-a)^(1/2)-2*a*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2
)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)))/sin(d*x+c)/(-(a*cos(d*x+c)^2-a-
b)/(cos(d*x+c)+1)^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((a+b*csc(d*x+c)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.07159, size = 3241, normalized size = 40.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*c
os(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c
)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 -
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*
sin(d*x + c)) + 2*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 +
4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1
))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)))/d, -1/8*(4*sqrt(-b)*arctan(-1/2
*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/
(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c))) - sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos
(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a
^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5
+ 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*c
os(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)))/d, 1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 -
8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1
))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))
) + sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*co
s(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c
) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)))/d, 1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c
)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^
2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x
+ c))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos
(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c))))/d]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*csc(d*x + c)^2 + a), x)