3.1 \(\int \frac{\tan ^4(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=55 \[ \frac{x}{a}-\frac{\tan ^5(x) (1-\csc (x))}{5 a}+\frac{\tan ^3(x) (5-4 \csc (x))}{15 a}-\frac{\tan (x) (15-8 \csc (x))}{15 a} \]

[Out]

x/a - ((15 - 8*Csc[x])*Tan[x])/(15*a) + ((5 - 4*Csc[x])*Tan[x]^3)/(15*a) - ((1 - Csc[x])*Tan[x]^5)/(5*a)

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Rubi [A]  time = 0.102983, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac{x}{a}-\frac{\tan ^5(x) (1-\csc (x))}{5 a}+\frac{\tan ^3(x) (5-4 \csc (x))}{15 a}-\frac{\tan (x) (15-8 \csc (x))}{15 a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^4/(a + a*Csc[x]),x]

[Out]

x/a - ((15 - 8*Csc[x])*Tan[x])/(15*a) + ((5 - 4*Csc[x])*Tan[x]^3)/(15*a) - ((1 - Csc[x])*Tan[x]^5)/(5*a)

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\tan ^4(x)}{a+a \csc (x)} \, dx &=\frac{\int (-a+a \csc (x)) \tan ^6(x) \, dx}{a^2}\\ &=-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int (5 a-4 a \csc (x)) \tan ^4(x) \, dx}{5 a^2}\\ &=\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int (-15 a+8 a \csc (x)) \tan ^2(x) \, dx}{15 a^2}\\ &=-\frac{(15-8 \csc (x)) \tan (x)}{15 a}+\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int 15 a \, dx}{15 a^2}\\ &=\frac{x}{a}-\frac{(15-8 \csc (x)) \tan (x)}{15 a}+\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}\\ \end{align*}

Mathematica [B]  time = 0.148421, size = 111, normalized size = 2.02 \[ \frac{-64 \sin (x)+240 x \sin (2 x)-178 \sin (2 x)-128 \sin (3 x)+120 x \sin (4 x)-89 \sin (4 x)+6 (120 x-89) \cos (x)+128 \cos (2 x)+240 x \cos (3 x)-178 \cos (3 x)+184 \cos (4 x)+200}{960 a \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^3 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^4/(a + a*Csc[x]),x]

[Out]

(200 + 6*(-89 + 120*x)*Cos[x] + 128*Cos[2*x] - 178*Cos[3*x] + 240*x*Cos[3*x] + 184*Cos[4*x] - 64*Sin[x] - 178*
Sin[2*x] + 240*x*Sin[2*x] - 128*Sin[3*x] - 89*Sin[4*x] + 120*x*Sin[4*x])/(960*a*(Cos[x/2] - Sin[x/2])^3*(Cos[x
/2] + Sin[x/2])^5)

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Maple [B]  time = 0.053, size = 102, normalized size = 1.9 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+{\frac{2}{5\,a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}+{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{11}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{6\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{4\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{5}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^4/(a+a*csc(x)),x)

[Out]

2/a*arctan(tan(1/2*x))+2/5/a/(tan(1/2*x)+1)^5-1/a/(tan(1/2*x)+1)^4+1/a/(tan(1/2*x)+1)^2+11/8/a/(tan(1/2*x)+1)-
1/6/a/(tan(1/2*x)-1)^3-1/4/a/(tan(1/2*x)-1)^2+5/8/a/(tan(1/2*x)-1)

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Maxima [B]  time = 1.48135, size = 262, normalized size = 4.76 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{46 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{13 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{100 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{35 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac{30 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{15 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + 8\right )}}{15 \,{\left (a + \frac{2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{6 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{6 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{2 \, a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{2 \, a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac{a \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} + \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^4/(a+a*csc(x)),x, algorithm="maxima")

[Out]

2/15*(sin(x)/(cos(x) + 1) - 46*sin(x)^2/(cos(x) + 1)^2 - 13*sin(x)^3/(cos(x) + 1)^3 + 100*sin(x)^4/(cos(x) + 1
)^4 + 35*sin(x)^5/(cos(x) + 1)^5 - 30*sin(x)^6/(cos(x) + 1)^6 - 15*sin(x)^7/(cos(x) + 1)^7 + 8)/(a + 2*a*sin(x
)/(cos(x) + 1) - 2*a*sin(x)^2/(cos(x) + 1)^2 - 6*a*sin(x)^3/(cos(x) + 1)^3 + 6*a*sin(x)^5/(cos(x) + 1)^5 + 2*a
*sin(x)^6/(cos(x) + 1)^6 - 2*a*sin(x)^7/(cos(x) + 1)^7 - a*sin(x)^8/(cos(x) + 1)^8) + 2*arctan(sin(x)/(cos(x)
+ 1))/a

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Fricas [A]  time = 0.488463, size = 174, normalized size = 3.16 \begin{align*} \frac{15 \, x \cos \left (x\right )^{3} + 23 \, \cos \left (x\right )^{4} - 19 \, \cos \left (x\right )^{2} +{\left (15 \, x \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) + 4}{15 \,{\left (a \cos \left (x\right )^{3} \sin \left (x\right ) + a \cos \left (x\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^4/(a+a*csc(x)),x, algorithm="fricas")

[Out]

1/15*(15*x*cos(x)^3 + 23*cos(x)^4 - 19*cos(x)^2 + (15*x*cos(x)^3 - 8*cos(x)^2 + 1)*sin(x) + 4)/(a*cos(x)^3*sin
(x) + a*cos(x)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{4}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**4/(a+a*csc(x)),x)

[Out]

Integral(tan(x)**4/(csc(x) + 1), x)/a

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Giac [A]  time = 1.40345, size = 108, normalized size = 1.96 \begin{align*} \frac{x}{a} + \frac{15 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 36 \, \tan \left (\frac{1}{2} \, x\right ) + 17}{24 \, a{\left (\tan \left (\frac{1}{2} \, x\right ) - 1\right )}^{3}} + \frac{55 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 260 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 450 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 300 \, \tan \left (\frac{1}{2} \, x\right ) + 71}{40 \, a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^4/(a+a*csc(x)),x, algorithm="giac")

[Out]

x/a + 1/24*(15*tan(1/2*x)^2 - 36*tan(1/2*x) + 17)/(a*(tan(1/2*x) - 1)^3) + 1/40*(55*tan(1/2*x)^4 + 260*tan(1/2
*x)^3 + 450*tan(1/2*x)^2 + 300*tan(1/2*x) + 71)/(a*(tan(1/2*x) + 1)^5)