3.891 \(\int \sin ^3(5 x) \tan ^4(5 x) \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{15} \cos ^3(5 x)-\frac {3}{5} \cos (5 x)+\frac {1}{15} \sec ^3(5 x)-\frac {3}{5} \sec (5 x) \]

[Out]

-3/5*cos(5*x)+1/15*cos(5*x)^3-3/5*sec(5*x)+1/15*sec(5*x)^3

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2590, 270} \[ \frac {1}{15} \cos ^3(5 x)-\frac {3}{5} \cos (5 x)+\frac {1}{15} \sec ^3(5 x)-\frac {3}{5} \sec (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[5*x]^3*Tan[5*x]^4,x]

[Out]

(-3*Cos[5*x])/5 + Cos[5*x]^3/15 - (3*Sec[5*x])/5 + Sec[5*x]^3/15

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rubi steps

\begin {align*} \int \sin ^3(5 x) \tan ^4(5 x) \, dx &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (5 x)\right )\right )\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \left (3+\frac {1}{x^4}-\frac {3}{x^2}-x^2\right ) \, dx,x,\cos (5 x)\right )\right )\\ &=-\frac {3}{5} \cos (5 x)+\frac {1}{15} \cos ^3(5 x)-\frac {3}{5} \sec (5 x)+\frac {1}{15} \sec ^3(5 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 0.95 \[ -\frac {11}{20} \cos (5 x)+\frac {1}{60} \cos (15 x)+\frac {1}{15} \sec ^3(5 x)-\frac {3}{5} \sec (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[5*x]^3*Tan[5*x]^4,x]

[Out]

(-11*Cos[5*x])/20 + Cos[15*x]/60 - (3*Sec[5*x])/5 + Sec[5*x]^3/15

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fricas [A]  time = 1.85, size = 32, normalized size = 0.86 \[ \frac {\cos \left (5 \, x\right )^{6} - 9 \, \cos \left (5 \, x\right )^{4} - 9 \, \cos \left (5 \, x\right )^{2} + 1}{15 \, \cos \left (5 \, x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)^3*tan(5*x)^4,x, algorithm="fricas")

[Out]

1/15*(cos(5*x)^6 - 9*cos(5*x)^4 - 9*cos(5*x)^2 + 1)/cos(5*x)^3

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giac [A]  time = 0.76, size = 33, normalized size = 0.89 \[ \frac {1}{15} \, \cos \left (5 \, x\right )^{3} - \frac {9 \, \cos \left (5 \, x\right )^{2} - 1}{15 \, \cos \left (5 \, x\right )^{3}} - \frac {3}{5} \, \cos \left (5 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)^3*tan(5*x)^4,x, algorithm="giac")

[Out]

1/15*cos(5*x)^3 - 1/15*(9*cos(5*x)^2 - 1)/cos(5*x)^3 - 3/5*cos(5*x)

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maple [B]  time = 0.11, size = 60, normalized size = 1.62 \[ \frac {\sin ^{8}\left (5 x \right )}{15 \cos \left (5 x \right )^{3}}-\frac {\sin ^{8}\left (5 x \right )}{3 \cos \left (5 x \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (5 x \right )+\frac {6 \left (\sin ^{4}\left (5 x \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (5 x \right )\right )}{5}\right ) \cos \left (5 x \right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(5*x)^3*tan(5*x)^4,x)

[Out]

1/15*sin(5*x)^8/cos(5*x)^3-1/3*sin(5*x)^8/cos(5*x)-1/3*(16/5+sin(5*x)^6+6/5*sin(5*x)^4+8/5*sin(5*x)^2)*cos(5*x
)

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maxima [A]  time = 0.31, size = 33, normalized size = 0.89 \[ \frac {1}{15} \, \cos \left (5 \, x\right )^{3} - \frac {9 \, \cos \left (5 \, x\right )^{2} - 1}{15 \, \cos \left (5 \, x\right )^{3}} - \frac {3}{5} \, \cos \left (5 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)^3*tan(5*x)^4,x, algorithm="maxima")

[Out]

1/15*cos(5*x)^3 - 1/15*(9*cos(5*x)^2 - 1)/cos(5*x)^3 - 3/5*cos(5*x)

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mupad [B]  time = 3.10, size = 30, normalized size = 0.81 \[ \frac {{\left (\cos \left (5\,x\right )+1\right )}^4\,\left ({\cos \left (5\,x\right )}^2-4\,\cos \left (5\,x\right )+1\right )}{15\,{\cos \left (5\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(5*x)^3*tan(5*x)^4,x)

[Out]

((cos(5*x) + 1)^4*(cos(5*x)^2 - 4*cos(5*x) + 1))/(15*cos(5*x)^3)

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sympy [A]  time = 0.09, size = 34, normalized size = 0.92 \[ \frac {1 - 9 \cos ^{2}{\left (5 x \right )}}{15 \cos ^{3}{\left (5 x \right )}} + \frac {\cos ^{3}{\left (5 x \right )}}{15} - \frac {3 \cos {\left (5 x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)**3*tan(5*x)**4,x)

[Out]

(1 - 9*cos(5*x)**2)/(15*cos(5*x)**3) + cos(5*x)**3/15 - 3*cos(5*x)/5

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