3.12 \(\int \csc ^5(e+f x) (-4+3 \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=18 \[ \frac{\cot (e+f x) \csc ^3(e+f x)}{f} \]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^3)/f

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Rubi [A]  time = 0.0220708, 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} \[ \frac{\cot (e+f x) \csc ^3(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^5*(-4 + 3*Sin[e + f*x]^2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^3)/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 ^5(e+f x) \left (-4+3 \sin ^2(e+f x)\right ) \, dx &=\frac{\cot (e+f x) \csc ^3(e+f x)}{f}\\ \end{align*}

Mathematica [B]  time = 0.0318164, size = 39, normalized size = 2.17 \[ \frac{\csc ^4\left (\frac{1}{2} (e+f x)\right )}{16 f}-\frac{\sec ^4\left (\frac{1}{2} (e+f x)\right )}{16 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^5*(-4 + 3*Sin[e + f*x]^2),x]

[Out]

Csc[(e + f*x)/2]^4/(16*f) - Sec[(e + f*x)/2]^4/(16*f)

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Maple [B]  time = 0.076, size = 47, normalized size = 2.6 \begin{align*}{\frac{1}{f} \left ( -4\, \left ( -1/4\, \left ( \csc \left ( fx+e \right ) \right ) ^{3}-3/8\,\csc \left ( fx+e \right ) \right ) \cot \left ( fx+e \right ) -{\frac{3\,\csc \left ( fx+e \right ) \cot \left ( fx+e \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^5*(-4+3*sin(f*x+e)^2),x)

[Out]

1/f*(-4*(-1/4*csc(f*x+e)^3-3/8*csc(f*x+e))*cot(f*x+e)-3/2*csc(f*x+e)*cot(f*x+e))

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Maxima [A]  time = 0.946263, size = 43, normalized size = 2.39 \begin{align*} \frac{\cos \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(-4+3*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

cos(f*x + e)/((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*f)

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Fricas [A]  time = 1.54045, size = 78, normalized size = 4.33 \begin{align*} \frac{\cos \left (f x + e\right )}{f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(-4+3*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

cos(f*x + e)/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)

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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)**5*(-4+3*sin(f*x+e)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.16458, size = 132, normalized size = 7.33 \begin{align*} -\frac{\frac{{\left (\frac{2 \,{\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - 1\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac{2 \,{\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac{{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{16 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(-4+3*sin(f*x+e)^2),x, algorithm="giac")

[Out]

-1/16*((2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 1)*(cos(f*x + e) + 1)^2/(cos(f*x + e) - 1)^2 - 2*(cos(f*x +
e) - 1)/(cos(f*x + e) + 1) + (cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/f