3.354 \(\int \frac{\cos ^3(e+f x)}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{b^{3/2} f} \]

[Out]

-(ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(3/2)*f)) + ((a + b)*Sin[e + f*x])/(a*b*f*Sqrt
[a + b*Sin[e + f*x]^2])

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Rubi [A]  time = 0.10049, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3190, 385, 217, 206} \[ \frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{b^{3/2} f} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

-(ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(3/2)*f)) + ((a + b)*Sin[e + f*x])/(a*b*f*Sqrt
[a + b*Sin[e + f*x]^2])

Rule 3190

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

Rule 385

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

Rubi steps

\begin{align*} \int \frac{\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{b f}\\ &=\frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{b^{3/2} f}+\frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.178865, size = 88, normalized size = 1.17 \[ \frac{\sqrt{b} (a+b) \sin (e+f x)-a^{3/2} \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right )}{a b^{3/2} f \sqrt{a+b \sin ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

(Sqrt[b]*(a + b)*Sin[e + f*x] - a^(3/2)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sin[e + f*x]^2)/a]
)/(a*b^(3/2)*f*Sqrt[a + b*Sin[e + f*x]^2])

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Maple [A]  time = 0.946, size = 90, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( fx+e \right ) }{bf}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}}-{\frac{1}{f}\ln \left ( \sin \left ( fx+e \right ) \sqrt{b}+\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \right ){b}^{-{\frac{3}{2}}}}+{\frac{\sin \left ( fx+e \right ) }{af}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x)

[Out]

1/f*sin(f*x+e)/b/(a+b*sin(f*x+e)^2)^(1/2)-1/f/b^(3/2)*ln(sin(f*x+e)*b^(1/2)+(a+b*sin(f*x+e)^2)^(1/2))+sin(f*x+
e)/a/f/(a+b*sin(f*x+e)^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(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.04989, size = 1324, normalized size = 17.65 \begin{align*} \left [\frac{{\left (a b \cos \left (f x + e\right )^{2} - a^{2} - a b\right )} \sqrt{b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \,{\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \,{\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \,{\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} + 8 \,{\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \,{\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \,{\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{b} \sin \left (f x + e\right )\right ) - 8 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a b + b^{2}\right )} \sin \left (f x + e\right )}{8 \,{\left (a b^{3} f \cos \left (f x + e\right )^{2} -{\left (a^{2} b^{2} + a b^{3}\right )} f\right )}}, \frac{{\left (a b \cos \left (f x + e\right )^{2} - a^{2} - a b\right )} \sqrt{-b} \arctan \left (\frac{{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \,{\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-b}}{4 \,{\left (2 \, b^{3} \cos \left (f x + e\right )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} -{\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) - 4 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a b + b^{2}\right )} \sin \left (f x + e\right )}{4 \,{\left (a b^{3} f \cos \left (f x + e\right )^{2} -{\left (a^{2} b^{2} + a b^{3}\right )} f\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

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

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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(cos(f*x+e)**3/(a+b*sin(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.37571, size = 96, normalized size = 1.28 \begin{align*} \frac{\frac{\log \left ({\left | -\sqrt{b} \sin \left (f x + e\right ) + \sqrt{b \sin \left (f x + e\right )^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} + \frac{{\left (a + b\right )} \sin \left (f x + e\right )}{\sqrt{b \sin \left (f x + e\right )^{2} + a} a b}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

(log(abs(-sqrt(b)*sin(f*x + e) + sqrt(b*sin(f*x + e)^2 + a)))/b^(3/2) + (a + b)*sin(f*x + e)/(sqrt(b*sin(f*x +
e)^2 + a)*a*b))/f