3.11 \(\int \csc ^4(e+f x) (-3+2 \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=18 \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/f

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Rubi [A]  time = 0.0215144, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3011} \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^4*(-3 + 2*Sin[e + f*x]^2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/f

Rule 3011

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

\begin{align*} \int \csc ^4(e+f x) \left (-3+2 \sin ^2(e+f x)\right ) \, dx &=\frac{\cot (e+f x) \csc ^2(e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0445803, size = 18, normalized size = 1. \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^4*(-3 + 2*Sin[e + f*x]^2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/f

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Maple [A]  time = 0.049, size = 34, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ( -3\, \left ( -2/3-1/3\, \left ( \csc \left ( fx+e \right ) \right ) ^{2} \right ) \cot \left ( fx+e \right ) -2\,\cot \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x)

[Out]

1/f*(-3*(-2/3-1/3*csc(f*x+e)^2)*cot(f*x+e)-2*cot(f*x+e))

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Maxima [A]  time = 0.966618, size = 30, normalized size = 1.67 \begin{align*} \frac{\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

(tan(f*x + e)^2 + 1)/(f*tan(f*x + e)^3)

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Fricas [A]  time = 1.60481, size = 72, normalized size = 4. \begin{align*} -\frac{\cos \left (f x + e\right )}{{\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

-cos(f*x + e)/((f*cos(f*x + e)^2 - f)*sin(f*x + e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**4*(-3+2*sin(f*x+e)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.12219, size = 32, normalized size = 1.78 \begin{align*} \frac{\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="giac")

[Out]

(tan(f*x + e)^2 + 1)/(f*tan(f*x + e)^3)