3.4.55 \(\int \frac {\cos (e+f x)}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\) [355]

Optimal. Leaf size=29 \[ \frac {\sin (e+f x)}{a f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

sin(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3269, 197} \begin {gather*} \frac {\sin (e+f x)}{a f \sqrt {a+b \sin ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 3269

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {\sin (e+f x)}{a f \sqrt {a+b \sin ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 28, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {\sin \left (f x +e \right )}{a f \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}\)	\(28\)
default	\(\frac {\sin \left (f x +e \right )}{a f \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(1/2)

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Maxima [A]
time = 0.27, size = 29, normalized size = 1.00 \begin {gather*} \frac {\sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(f*x + e)/(sqrt(b*sin(f*x + e)^2 + a)*a*f)

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Fricas [A]
time = 0.43, size = 49, normalized size = 1.69 \begin {gather*} -\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{a b f \cos \left (f x + e\right )^{2} - {\left (a^{2} + a b\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(a*b*f*cos(f*x + e)^2 - (a^2 + a*b)*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.66, size = 29, normalized size = 1.00 \begin {gather*} \frac {\sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(f*x + e)/(sqrt(b*sin(f*x + e)^2 + a)*a*f)

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Mupad [B]
time = 15.61, size = 117, normalized size = 4.03 \begin {gather*} \frac {\sqrt {2}\,\sqrt {2\,a+b-b\,\cos \left (2\,e+2\,f\,x\right )}\,\left (4\,a\,\sin \left (e+f\,x\right )+3\,b\,\sin \left (e+f\,x\right )-b\,\sin \left (3\,e+3\,f\,x\right )\right )}{a\,f\,\left (8\,a\,b+8\,a^2+3\,b^2-4\,b^2\,\cos \left (2\,e+2\,f\,x\right )+b^2\,\cos \left (4\,e+4\,f\,x\right )-8\,a\,b\,\cos \left (2\,e+2\,f\,x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*(2*a + b - b*cos(2*e + 2*f*x))^(1/2)*(4*a*sin(e + f*x) + 3*b*sin(e + f*x) - b*sin(3*e + 3*f*x)))/(a*f
*(8*a*b + 8*a^2 + 3*b^2 - 4*b^2*cos(2*e + 2*f*x) + b^2*cos(4*e + 4*f*x) - 8*a*b*cos(2*e + 2*f*x)))

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