\(\int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\) [268]

Optimal result

Integrand size = 21, antiderivative size = 142 \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3} \]

[Out]

-2*cos(b*x+a)^5/b/d/(d*tan(b*x+a))^(1/2)+77/20*cos(b*x+a)*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*Ellipt
icE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*tan(b*x+a))^(1/2)/b/d^2/sin(2*b*x+2*a)^(1/2)-77/30*cos(b*x+a)^3*(d*tan(b*x+a
))^(3/2)/b/d^3-11/5*cos(b*x+a)^5*(d*tan(b*x+a))^(3/2)/b/d^3

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2689, 2692, 2695, 2652, 2719} \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {77 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}} \]

[In]

Int[Cos[a + b*x]^5/(d*Tan[a + b*x])^(3/2),x]

[Out]

(-2*Cos[a + b*x]^5)/(b*d*Sqrt[d*Tan[a + b*x]]) - (77*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a +
b*x]])/(20*b*d^2*Sqrt[Sin[2*a + 2*b*x]]) - (77*Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(30*b*d^3) - (11*Cos[a +
b*x]^5*(d*Tan[a + b*x])^(3/2))/(5*b*d^3)

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 2689

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*(n + 1))), x] - Dist[(m + n + 1)/(b^2*(n + 1)), Int[(a*Sec[e + f*x])^m*(
b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && IntegersQ[2*m, 2*n]

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*} \text {integral}& = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {11 \int \cos ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx}{d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \int \cos ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx}{10 d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \int \cos (a+b x) \sqrt {d \tan (a+b x)} \, dx}{20 d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {\left (77 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{20 d^2 \sqrt {\sin (a+b x)}} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {\left (77 \cos (a+b x) \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{20 d^2 \sqrt {\sin (2 a+2 b x)}} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.91 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sin (a+b x) \left (-277+34 \cos (2 (a+b x))+3 \cos (4 (a+b x))-308 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \tan ^2(a+b x)\right )}{120 b (d \tan (a+b x))^{3/2}} \]

[In]

Integrate[Cos[a + b*x]^5/(d*Tan[a + b*x])^(3/2),x]

[Out]

(Sin[a + b*x]*(-277 + 34*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)] - 308*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a +
b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(120*b*(d*Tan[a + b*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs. \(2(151)=302\).

Time = 1.39 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.77

method	result	size
default	\(\frac {\left (12 \left (\cos ^{5}\left (b x +a \right )\right ) \sqrt {2}+22 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+462 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-231 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+462 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-231 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+77 \sqrt {2}\, \cos \left (b x +a \right )-231 \sqrt {2}\right ) \sqrt {2}}{120 b \sqrt {d \tan \left (b x +a \right )}\, d}\)	\(393\)

[In]

int(cos(b*x+a)^5/(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/120/b/(d*tan(b*x+a))^(1/2)/d*(12*cos(b*x+a)^5*2^(1/2)+22*cos(b*x+a)^3*2^(1/2)+462*(1+csc(b*x+a)-cot(b*x+a))^
(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2)
,1/2*2^(1/2))-231*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1/
2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+462*sec(b*x+a)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc
(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2
))-231*sec(b*x+a)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/
2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+77*2^(1/2)*cos(b*x+a)-231*2^(1/2))*2^(1/2)

Fricas [F]

\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x+a)^5/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*tan(b*x + a))*cos(b*x + a)^5/(d^2*tan(b*x + a)^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(b*x+a)**5/(d*tan(b*x+a))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x+a)^5/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^5/(d*tan(b*x + a))^(3/2), x)

Giac [F]

\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x+a)^5/(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^5/(d*tan(b*x + a))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^5}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(cos(a + b*x)^5/(d*tan(a + b*x))^(3/2),x)

[Out]

int(cos(a + b*x)^5/(d*tan(a + b*x))^(3/2), x)