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

Optimal. Leaf size=16 \[ \frac{\cot (e+f x) \csc (e+f x)}{f} \]

[Out]

(Cot[e + f*x]*Csc[e + f*x])/f

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Rubi [A]  time = 0.0216889, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3011} \[ \frac{\cot (e+f x) \csc (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0262604, size = 107, normalized size = 6.69 \[ \frac{\csc ^2\left (\frac{1}{2} (e+f x)\right )}{4 f}-\frac{\sec ^2\left (\frac{1}{2} (e+f x)\right )}{4 f}+\frac{\log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}-\frac{\log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f}-\frac{\log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}+\frac{\log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[(e + f*x)/2]^2/(4*f) - Log[Cos[e/2 + (f*x)/2]]/f + Log[Cos[(e + f*x)/2]]/f + Log[Sin[e/2 + (f*x)/2]]/f - L
og[Sin[(e + f*x)/2]]/f - Sec[(e + f*x)/2]^2/(4*f)

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Maple [A]  time = 0.07, size = 17, normalized size = 1.1 \begin{align*}{\frac{\csc \left ( fx+e \right ) \cot \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cot(f*x+e)*csc(f*x+e)/f

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Maxima [A]  time = 0.980448, size = 31, normalized size = 1.94 \begin{align*} -\frac{\cos \left (f x + e\right )}{{\left (\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)^3*(-2+sin(f*x+e)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52423, size = 51, normalized size = 3.19 \begin{align*} -\frac{\cos \left (f x + e\right )}{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)^3*(-2+sin(f*x+e)^2),x, algorithm="fricas")

[Out]

-cos(f*x + e)/(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)**3*(-2+sin(f*x+e)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.14988, size = 66, normalized size = 4.12 \begin{align*} -\frac{\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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