3.39 \(\int (-\cot (x)+\tan (x))^2 \, dx\)

Optimal. Leaf size=10 \[ -4 x+\tan (x)-\cot (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {461, 203} \[ -4 x+\tan (x)-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[(-Cot[x] + Tan[x])^2,x]

[Out]

-4*x - Cot[x] + Tan[x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rubi steps

\begin {align*} \int (-\cot (x)+\tan (x))^2 \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2 \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {4}{1+x^2}\right ) \, dx,x,\tan (x)\right )\\ &=-\cot (x)+\tan (x)-4 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=-4 x-\cot (x)+\tan (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 10, normalized size = 1.00 \[ -4 x+\tan (x)-\cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cot[x] + Tan[x])^2,x]

[Out]

-4*x - Cot[x] + Tan[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int (-\cot (x)+\tan (x))^2 \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(-Cot[x] + Tan[x])^2,x]

[Out]

Could not integrate

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fricas [A]  time = 0.90, size = 19, normalized size = 1.90 \[ -\frac {4 \, x \tan \relax (x) - \tan \relax (x)^{2} + 1}{\tan \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cot(x)+tan(x))^2,x, algorithm="fricas")

[Out]

-(4*x*tan(x) - tan(x)^2 + 1)/tan(x)

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giac [A]  time = 0.83, size = 12, normalized size = 1.20 \[ -4 \, x - \frac {1}{\tan \relax (x)} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cot(x)+tan(x))^2,x, algorithm="giac")

[Out]

-4*x - 1/tan(x) + tan(x)

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maple [A]  time = 0.05, size = 11, normalized size = 1.10

method	result	size
default	\(-4 x -\cot \relax (x )+\tan \relax (x )\)	\(11\)
norman	\(\frac {-1+\tan ^{2}\relax (x )-4 x \tan \relax (x )}{\tan \relax (x )}\)	\(17\)
risch	\(-4 x -\frac {4 i}{\left (1+{\mathrm e}^{2 i x}\right ) \left ({\mathrm e}^{2 i x}-1\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-cot(x)+tan(x))^2,x,method=_RETURNVERBOSE)

[Out]

-4*x-cot(x)+tan(x)

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maxima [A]  time = 0.95, size = 12, normalized size = 1.20 \[ -4 \, x - \frac {1}{\tan \relax (x)} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cot(x)+tan(x))^2,x, algorithm="maxima")

[Out]

-4*x - 1/tan(x) + tan(x)

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mupad [B]  time = 0.27, size = 12, normalized size = 1.20 \[ \mathrm {tan}\relax (x)-4\,x-\frac {1}{\mathrm {tan}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x) - tan(x))^2,x)

[Out]

tan(x) - 4*x - 1/tan(x)

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sympy [A]  time = 0.37, size = 10, normalized size = 1.00 \[ - 4 x + \tan {\relax (x )} - \frac {1}{\tan {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cot(x)+tan(x))**2,x)

[Out]

-4*x + tan(x) - 1/tan(x)

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