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

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.03, 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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fricas [A]  time = 0.40, size = 24, normalized size = 1.50 \[ -\frac {\cos \left (f x + e\right )}{f \cos \left (f x + e\right )^{2} - f} \]

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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giac [B]  time = 0.18, size = 49, normalized size = 3.06 \[ -\frac {\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}}{4 \, f} \]

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

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maple [A]  time = 0.51, size = 17, normalized size = 1.06 \[ \frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )}{f} \]

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 = 0.33, size = 23, normalized size = 1.44 \[ -\frac {\cos \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} f} \]

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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mupad [B]  time = 13.23, size = 23, normalized size = 1.44 \[ -\frac {\cos \left (e+f\,x\right )}{f\,\left ({\cos \left (e+f\,x\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{2}{\left (e + f x \right )} - 2\right ) \csc ^{3}{\left (e + f x \right )}\, dx \]

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]

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

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