3.1.12 \(\int \csc ^5(e+f x) (-4+3 \sin ^2(e+f x)) \, dx\) [12]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {3090} \begin {gather*} \frac {\cot (e+f x) \csc ^3(e+f x)}{f} \end {gather*}

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 3090

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).
time = 0.03, size = 39, normalized size = 2.17 \begin {gather*} \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} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(18)=36\).
time = 0.29, size = 47, normalized size = 2.61

method	result	size
risch	\(\frac {8 \,{\mathrm e}^{5 i \left (f x +e \right )}+8 \,{\mathrm e}^{3 i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}\)	\(38\)
derivativedivides	\(\frac {-4 \left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )-\frac {3 \csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}}{f}\)	\(47\)
default	\(\frac {-4 \left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )-\frac {3 \csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}}{f}\)	\(47\)
norman	\(\frac {\frac {1}{16 f}+\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )}{16 f}+\frac {5 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {5 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\)	\(114\)

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,method=_RETURNVERBOSE)

[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 = 0.36, size = 35, normalized size = 1.94 \begin {gather*} \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 {gather*}

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 = 0.73, size = 35, normalized size = 1.94 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (3 \sin ^{2}{\left (e + f x \right )} - 4\right ) \csc ^{5}{\left (e + f x \right )}\, dx \end {gather*}

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (20) = 40\).
time = 0.61, size = 98, normalized size = 5.44 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 13.23, size = 22, normalized size = 1.22 \begin {gather*} \frac {\cos \left (e+f\,x\right )}{f\,{\left ({\cos \left (e+f\,x\right )}^2-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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