3.60 \(\int \frac{\csc ^4(c+d x)}{(a-a \sin ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=65 \[ \frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{3 \tan (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d} \]

[Out]

(-3*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)

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Rubi [A]  time = 0.0808019, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2620, 270} \[ \frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{3 \tan (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]

[Out]

(-3*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 2620

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

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]

Rubi steps

\begin{align*} \int \frac{\csc ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \csc ^4(c+d x) \sec ^4(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^4} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}+\frac{3}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac{3 \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{3 \tan (c+d x)}{a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0246865, size = 46, normalized size = 0.71 \[ \frac{16 \left (-\frac{\cot (2 (c+d x))}{3 d}-\frac{\cot (2 (c+d x)) \csc ^2(2 (c+d x))}{6 d}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]

[Out]

(16*(-Cot[2*(c + d*x)]/(3*d) - (Cot[2*(c + d*x)]*Csc[2*(c + d*x)]^2)/(6*d)))/a^2

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Maple [A]  time = 0.076, size = 47, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}d} \left ({\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}+3\,\tan \left ( dx+c \right ) -3\, \left ( \tan \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^4/(a-sin(d*x+c)^2*a)^2,x)

[Out]

1/d/a^2*(1/3*tan(d*x+c)^3+3*tan(d*x+c)-3/tan(d*x+c)-1/3/tan(d*x+c)^3)

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Maxima [A]  time = 0.955788, size = 70, normalized size = 1.08 \begin{align*} \frac{\frac{\tan \left (d x + c\right )^{3} + 9 \, \tan \left (d x + c\right )}{a^{2}} - \frac{9 \, \tan \left (d x + c\right )^{2} + 1}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/3*((tan(d*x + c)^3 + 9*tan(d*x + c))/a^2 - (9*tan(d*x + c)^2 + 1)/(a^2*tan(d*x + c)^3))/d

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Fricas [A]  time = 1.64036, size = 176, normalized size = 2.71 \begin{align*} -\frac{16 \, \cos \left (d x + c\right )^{6} - 24 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} + 1}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(16*cos(d*x + c)^6 - 24*cos(d*x + c)^4 + 6*cos(d*x + c)^2 + 1)/((a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c
)^3)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{4}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} - 2 \sin ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**4/(a-a*sin(d*x+c)**2)**2,x)

[Out]

Integral(csc(c + d*x)**4/(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1), x)/a**2

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Giac [A]  time = 1.16087, size = 46, normalized size = 0.71 \begin{align*} -\frac{8 \,{\left (3 \, \tan \left (2 \, d x + 2 \, c\right )^{2} + 1\right )}}{3 \, a^{2} d \tan \left (2 \, d x + 2 \, c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-8/3*(3*tan(2*d*x + 2*c)^2 + 1)/(a^2*d*tan(2*d*x + 2*c)^3)