3.3.35 \(\int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx\) [235]

Optimal. Leaf size=111 \[ \frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f} \]

[Out]

-7/20*cos(f*x+e)*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(d*tan(f*x
+e))^(1/2)/f/sin(2*f*x+2*e)^(1/2)+7/30*cos(f*x+e)^3*(d*tan(f*x+e))^(3/2)/d/f+1/5*cos(f*x+e)^5*(d*tan(f*x+e))^(
3/2)/d/f

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Rubi [A]
time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2692, 2695, 2652, 2719} \begin {gather*} \frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {7 \cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]],x]

[Out]

(7*Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(20*f*Sqrt[Sin[2*e + 2*f*x]]) + (7*Cos[e +
f*x]^3*(d*Tan[e + f*x])^(3/2))/(30*d*f) + (Cos[e + f*x]^5*(d*Tan[e + f*x])^(3/2))/(5*d*f)

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[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 2695

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

Rule 2719

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

Rubi steps

\begin {align*} \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx &=\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{10} \int \cos ^3(e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{20} \int \cos (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {\left (7 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{20 \sqrt {\sin (e+f x)}}\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {\left (7 \cos (e+f x) \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{20 \sqrt {\sin (2 e+2 f x)}}\\ &=\frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.83, size = 86, normalized size = 0.77 \begin {gather*} \frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \left (20 \sin (2 (e+f x))+3 \sin (4 (e+f x))+28 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x)} \tan (e+f x)\right )}{120 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]],x]

[Out]

(Cos[e + f*x]*Sqrt[d*Tan[e + f*x]]*(20*Sin[2*(e + f*x)] + 3*Sin[4*(e + f*x)] + 28*Hypergeometric2F1[3/4, 3/2,
7/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x]))/(120*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(549\) vs. \(2(122)=244\).
time = 0.35, size = 550, normalized size = 4.95

method	result	size
default	\(-\frac {\left (\cos \left (f x +e \right )-1\right )^{2} \left (12 \sqrt {2}\, \left (\cos ^{6}\left (f x +e \right )\right )+2 \sqrt {2}\, \left (\cos ^{4}\left (f x +e \right )\right )-21 \sqrt {\frac {\cos \left (f x +e \right )-1}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+42 \sqrt {\frac {\cos \left (f x +e \right )-1}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-21 \sqrt {\frac {\cos \left (f x +e \right )-1}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {\frac {\cos \left (f x +e \right )-1}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+7 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-21 \cos \left (f x +e \right ) \sqrt {2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {2}}{120 f \sin \left (f x +e \right )^{5}}\)	\(550\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/120/f*(cos(f*x+e)-1)^2*(12*2^(1/2)*cos(f*x+e)^6+2*2^(1/2)*cos(f*x+e)^4-21*((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)*(-(-1+cos(f*x+e)-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)+42*((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)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e
)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*cos(f*x+e)-21*((cos(f*x+e)-1)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+s
in(f*x+e))/sin(f*x+e))^(1/2)*(-(-1+cos(f*x+e)-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))+42*((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)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/
2),1/2*2^(1/2))+7*cos(f*x+e)^2*2^(1/2)-21*cos(f*x+e)*2^(1/2))*(cos(f*x+e)+1)^2*(d*sin(f*x+e)/cos(f*x+e))^(1/2)
/sin(f*x+e)^5*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**5*(d*tan(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (e+f\,x\right )}^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)^5*(d*tan(e + f*x))^(1/2),x)

[Out]

int(cos(e + f*x)^5*(d*tan(e + f*x))^(1/2), x)

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