3.346 \(\int \sin ^{10}(x) \tan (x) \, dx\)

Optimal. Leaf size=46 \[ \frac{\cos ^{10}(x)}{10}-\frac{5 \cos ^8(x)}{8}+\frac{5 \cos ^6(x)}{3}-\frac{5 \cos ^4(x)}{2}+\frac{5 \cos ^2(x)}{2}-\log (\cos (x)) \]

[Out]

(5*Cos[x]^2)/2 - (5*Cos[x]^4)/2 + (5*Cos[x]^6)/3 - (5*Cos[x]^8)/8 + Cos[x]^10/10 - Log[Cos[x]]

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Rubi [A]  time = 0.0280991, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2590, 266, 43} \[ \frac{\cos ^{10}(x)}{10}-\frac{5 \cos ^8(x)}{8}+\frac{5 \cos ^6(x)}{3}-\frac{5 \cos ^4(x)}{2}+\frac{5 \cos ^2(x)}{2}-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^10*Tan[x],x]

[Out]

(5*Cos[x]^2)/2 - (5*Cos[x]^4)/2 + (5*Cos[x]^6)/3 - (5*Cos[x]^8)/8 + Cos[x]^10/10 - Log[Cos[x]]

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]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \sin ^{10}(x) \tan (x) \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^5}{x} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1-x)^5}{x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-5+\frac{1}{x}+10 x-10 x^2+5 x^3-x^4\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac{5 \cos ^2(x)}{2}-\frac{5 \cos ^4(x)}{2}+\frac{5 \cos ^6(x)}{3}-\frac{5 \cos ^8(x)}{8}+\frac{\cos ^{10}(x)}{10}-\log (\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.0149302, size = 46, normalized size = 1. \[ \frac{\cos ^{10}(x)}{10}-\frac{5 \cos ^8(x)}{8}+\frac{5 \cos ^6(x)}{3}-\frac{5 \cos ^4(x)}{2}+\frac{5 \cos ^2(x)}{2}-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^10*Tan[x],x]

[Out]

(5*Cos[x]^2)/2 - (5*Cos[x]^4)/2 + (5*Cos[x]^6)/3 - (5*Cos[x]^8)/8 + Cos[x]^10/10 - Log[Cos[x]]

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Maple [A]  time = 0.01, size = 37, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{10}}{10}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{8}}{8}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{6}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}}-\ln \left ( \cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^11/cos(x),x)

[Out]

-1/10*sin(x)^10-1/8*sin(x)^8-1/6*sin(x)^6-1/4*sin(x)^4-1/2*sin(x)^2-ln(cos(x))

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Maxima [A]  time = 0.935469, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{10} \, \sin \left (x\right )^{10} - \frac{1}{8} \, \sin \left (x\right )^{8} - \frac{1}{6} \, \sin \left (x\right )^{6} - \frac{1}{4} \, \sin \left (x\right )^{4} - \frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^11/cos(x),x, algorithm="maxima")

[Out]

-1/10*sin(x)^10 - 1/8*sin(x)^8 - 1/6*sin(x)^6 - 1/4*sin(x)^4 - 1/2*sin(x)^2 - 1/2*log(sin(x)^2 - 1)

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Fricas [A]  time = 2.12827, size = 123, normalized size = 2.67 \begin{align*} \frac{1}{10} \, \cos \left (x\right )^{10} - \frac{5}{8} \, \cos \left (x\right )^{8} + \frac{5}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} - \log \left (-\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^11/cos(x),x, algorithm="fricas")

[Out]

1/10*cos(x)^10 - 5/8*cos(x)^8 + 5/3*cos(x)^6 - 5/2*cos(x)^4 + 5/2*cos(x)^2 - log(-cos(x))

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Sympy [A]  time = 0.090494, size = 44, normalized size = 0.96 \begin{align*} - \log{\left (\cos{\left (x \right )} \right )} + \frac{\cos ^{10}{\left (x \right )}}{10} - \frac{5 \cos ^{8}{\left (x \right )}}{8} + \frac{5 \cos ^{6}{\left (x \right )}}{3} - \frac{5 \cos ^{4}{\left (x \right )}}{2} + \frac{5 \cos ^{2}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**11/cos(x),x)

[Out]

-log(cos(x)) + cos(x)**10/10 - 5*cos(x)**8/8 + 5*cos(x)**6/3 - 5*cos(x)**4/2 + 5*cos(x)**2/2

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Giac [A]  time = 1.07015, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{10} \, \cos \left (x\right )^{10} - \frac{5}{8} \, \cos \left (x\right )^{8} + \frac{5}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} - \frac{1}{2} \, \log \left (\cos \left (x\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^11/cos(x),x, algorithm="giac")

[Out]

1/10*cos(x)^10 - 5/8*cos(x)^8 + 5/3*cos(x)^6 - 5/2*cos(x)^4 + 5/2*cos(x)^2 - 1/2*log(cos(x)^2)