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

Optimal. Leaf size=46 \[ \frac{\tan (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{2 \cot (c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0801195, antiderivative size = 46, 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} \[ \frac{\tan (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{2 \cot (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0430301, size = 49, normalized size = 1.07 \[ \frac{\frac{\tan (c+d x)}{d}-\frac{5 \cot (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*Cot[c + d*x])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) + Tan[c + d*x]/d)/a

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Maple [A]  time = 0.075, size = 35, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( \tan \left ( dx+c \right ) -2\, \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),x)

[Out]

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

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Maxima [A]  time = 0.955211, size = 57, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, \tan \left (d x + c\right )}{a} - \frac{6 \, \tan \left (d x + c\right )^{2} + 1}{a \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),x, algorithm="maxima")

[Out]

1/3*(3*tan(d*x + c)/a - (6*tan(d*x + c)^2 + 1)/(a*tan(d*x + c)^3))/d

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Fricas [A]  time = 1.62205, size = 140, normalized size = 3.04 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 3}{3 \,{\left (a d \cos \left (d x + c\right )^{3} - a d \cos \left (d x + c\right )\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),x, algorithm="fricas")

[Out]

-1/3*(8*cos(d*x + c)^4 - 12*cos(d*x + c)^2 + 3)/((a*d*cos(d*x + c)^3 - a*d*cos(d*x + c))*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 ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \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),x)

[Out]

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

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Giac [A]  time = 1.1818, size = 57, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, \tan \left (d x + c\right )}{a} - \frac{6 \, \tan \left (d x + c\right )^{2} + 1}{a \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),x, algorithm="giac")

[Out]

1/3*(3*tan(d*x + c)/a - (6*tan(d*x + c)^2 + 1)/(a*tan(d*x + c)^3))/d