3.256 \(\int \frac{\cos (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=76 \[ \frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 f \sqrt{d \tan (e+f x)}} \]

[Out]

(EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[d*Tan[e + f*x]]) + (Cos[e + f*x]*
Sqrt[d*Tan[e + f*x]])/(d*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0932574, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2612, 2614, 2573, 2641} \[ \frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 f \sqrt{d \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]/Sqrt[d*Tan[e + f*x]],x]

[Out]

(EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[d*Tan[e + f*x]]) + (Cos[e + f*x]*
Sqrt[d*Tan[e + f*x]])/(d*f)

Rule 2612

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] + Dist[(m + n + 1)/(a^2*m), Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[
e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, -2^(-1)])) && Integer
sQ[2*m, 2*n]

Rule 2614

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Co
s[e + f*x]]*Sqrt[b*Tan[e + f*x]]), Int[1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x
]

Rule 2573

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cos (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{1}{2} \int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (e+f x)} \int \frac{1}{\sqrt{\cos (e+f x)} \sqrt{\sin (e+f x)}} \, dx}{2 \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\left (\sec (e+f x) \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{2 \sqrt{d \tan (e+f x)}}\\ &=\frac{F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt{\sin (2 e+2 f x)}}{2 f \sqrt{d \tan (e+f x)}}+\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}\\ \end{align*}

Mathematica [C]  time = 0.537385, size = 126, normalized size = 1.66 \[ \frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \sec (e+f x) \left (-\sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)}+\sqrt [4]{-1} \sec ^2(e+f x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (e+f x)}\right )\right |-1\right )\right )}{f \left (\tan ^2(e+f x)-1\right ) \sqrt{\sec ^2(e+f x)} \sqrt{d \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]/Sqrt[d*Tan[e + f*x]],x]

[Out]

(Cos[2*(e + f*x)]*Sec[e + f*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sec[e + f*x
]^2 - Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])*Sqrt[Tan[e + f*x]])/(f*Sqrt[Sec[e + f*x]^2]*Sqrt[d*Tan[e + f*x]
]*(-1 + Tan[e + f*x]^2))

________________________________________________________________________________________

Maple [B]  time = 0.145, size = 196, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{2}+\cos \left ( fx+e \right ) \sqrt{2} \right ){\frac{1}{\sqrt{{\frac{d\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)/(d*tan(f*x+e))^(1/2),x)

[Out]

-1/2/f*2^(1/2)*(cos(f*x+e)-1)*(sin(f*x+e)*((cos(f*x+e)-1)/sin(f*x+e))^(1/2)*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x
+e))^(1/2)*((cos(f*x+e)-1+sin(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)
,1/2*2^(1/2))-cos(f*x+e)^2*2^(1/2)+cos(f*x+e)*2^(1/2))*(cos(f*x+e)+1)^2/sin(f*x+e)^3/cos(f*x+e)/(d*sin(f*x+e)/
cos(f*x+e))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (f x + e\right )}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(f*x + e)/sqrt(d*tan(f*x + e)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (f x + e\right )} \cos \left (f x + e\right )}{d \tan \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*tan(f*x + e))*cos(f*x + e)/(d*tan(f*x + e)), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (e + f x \right )}}{\sqrt{d \tan{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)/(d*tan(f*x+e))**(1/2),x)

[Out]

Integral(cos(e + f*x)/sqrt(d*tan(e + f*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (f x + e\right )}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(f*x + e)/sqrt(d*tan(f*x + e)), x)