3.81 \(\int \frac{\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{b} d \sqrt{a+b}} \]

[Out]

-(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d))

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Rubi [A]  time = 0.0395803, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3186, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{b} d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*Sin[c + d*x]^2),x]

[Out]

-(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d))

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

Mathematica [C]  time = 0.156883, size = 97, normalized size = 2.62 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )+\tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )}{\sqrt{b} d \sqrt{-a-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/S
qrt[-a - b]])/(Sqrt[-a - b]*Sqrt[b]*d)

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Maple [A]  time = 0.063, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{d}{\it Artanh} \left ({b\cos \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77946, size = 273, normalized size = 7.38 \begin{align*} \left [\frac{\log \left (-\frac{b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right )}{2 \, \sqrt{a b + b^{2}} d}, \frac{\sqrt{-a b - b^{2}} \arctan \left (\frac{\sqrt{-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right )}{{\left (a b + b^{2}\right )} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [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(sin(d*x+c)/(a+b*sin(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.1264, size = 50, normalized size = 1.35 \begin{align*} \frac{\arctan \left (\frac{b \cos \left (d x + c\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*d)