\(\int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\) [257]

Optimal result

Integrand size = 21, antiderivative size = 109 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {5 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {d \tan (e+f x)}}+\frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f} \]

[Out]

-5/12*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*sec(f*x+e)*sin(2*f*x+
2*e)^(1/2)/f/(d*tan(f*x+e))^(1/2)+5/6*cos(f*x+e)*(d*tan(f*x+e))^(1/2)/d/f+1/3*cos(f*x+e)^3*(d*tan(f*x+e))^(1/2
)/d/f

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2692, 2694, 2653, 2720} \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {5 \sqrt {\sin (2 e+2 f x)} \sec (e+f x) \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{12 f \sqrt {d \tan (e+f x)}} \]

[In]

Int[Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]

[Out]

(5*EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(12*f*Sqrt[d*Tan[e + f*x]]) + (5*Cos[e +
f*x]*Sqrt[d*Tan[e + f*x]])/(6*d*f) + (Cos[e + f*x]^3*Sqrt[d*Tan[e + f*x]])/(3*d*f)

Rule 2653

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 2692

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*Ta
n[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, -2^(-1)])) && Integ
ersQ[2*m, 2*n]

Rule 2694

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 2720

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

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {5}{6} \int \frac {\cos (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx \\ & = \frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {5}{12} \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx \\ & = \frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {\left (5 \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}} \, dx}{12 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}} \\ & = \frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {\left (5 \sec (e+f x) \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{12 \sqrt {d \tan (e+f x)}} \\ & = \frac {5 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {d \tan (e+f x)}}+\frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {11 \sin (e+f x)+\sin (3 (e+f x))-10 \sqrt [4]{-1} \cos (e+f x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (e+f x)}\right ),-1\right ) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}}{12 f \sqrt {d \tan (e+f x)}} \]

[In]

Integrate[Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]

[Out]

(11*Sin[e + f*x] + Sin[3*(e + f*x)] - 10*(-1)^(1/4)*Cos[e + f*x]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f
*x]]], -1]*Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])/(12*f*Sqrt[d*Tan[e + f*x]])

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.23 (sec) , antiderivative size = 1906, normalized size of antiderivative = 17.49

method	result	size
default	\(\text {Expression too large to display}\)	\(1906\)

[In]

int(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/48/f/(d*tan(f*x+e))^(1/2)*(6*I*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-c
ot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-6*I*sec(f*x+e)*(cot(f*x+
e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticPi((-cot(f*x+e)+
csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*I*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*(
-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*sec(f*x
+e)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),
1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)+6*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+
e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-
32*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticF((-
cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))+6*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*(-
cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-8*cos(f*x+e)
^2*sin(f*x+e)*2^(1/2)+6*sec(f*x+e)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-
cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-32*sec(f*x+e)*(-cot(f*x
+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+c
sc(f*x+e)+1)^(1/2),1/2*2^(1/2))+6*sec(f*x+e)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*(-c
ot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*ln(-(cot(f*
x+e)*cos(f*x+e)-2*cot(f*x+e)-2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+c
sc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))*(-sin(f*x+e)*co
s(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-3*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*ln(-(cot(f*x+e)*cos(f*x+e)-
2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(
f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))-6*arctan((2^(1/2)*(-sin(f*x+e)*
cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e
)+1)^2)^(1/2)-6*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(cos(
f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-20*2^(1/2)*sin(f*x+e)+3*sec(f*x+e)*ln(-(cot(f*x+e)*
cos(f*x+e)-2*cot(f*x+e)-2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*
x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x
+e)/(cos(f*x+e)+1)^2)^(1/2)-3*sec(f*x+e)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*ln(-(cot(f*x+e)*cos(f
*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^
3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))-6*sec(f*x+e)*arctan((2^(1
/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(
f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-6*sec(f*x+e)*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*si
n(f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2)

Fricas [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*tan(f*x + e))*cos(f*x + e)^3/(d*tan(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**3/(d*tan(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(f*x + e)^3/sqrt(d*tan(f*x + e)), x)

Giac [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(f*x + e)^3/sqrt(d*tan(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^3}{\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]

[In]

int(cos(e + f*x)^3/(d*tan(e + f*x))^(1/2),x)

[Out]

int(cos(e + f*x)^3/(d*tan(e + f*x))^(1/2), x)