3.167 \(\int \csc ^4(a+b x) \sec ^2(a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac {\tan (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}-\frac {2 \cot (a+b x)}{b} \]

[Out]

-2*cot(b*x+a)/b-1/3*cot(b*x+a)^3/b+tan(b*x+a)/b

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2620, 270} \[ \frac {\tan (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}-\frac {2 \cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4*Sec[a + b*x]^2,x]

[Out]

(-2*Cot[a + b*x])/b - Cot[a + b*x]^3/(3*b) + Tan[a + b*x]/b

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rubi steps

\begin {align*} \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac {2 \cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 45, normalized size = 1.22 \[ \frac {\tan (a+b x)}{b}-\frac {5 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4*Sec[a + b*x]^2,x]

[Out]

(-5*Cot[a + b*x])/(3*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(3*b) + Tan[a + b*x]/b

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fricas [A]  time = 0.44, size = 54, normalized size = 1.46 \[ -\frac {8 \, \cos \left (b x + a\right )^{4} - 12 \, \cos \left (b x + a\right )^{2} + 3}{3 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2/sin(b*x+a)^4,x, algorithm="fricas")

[Out]

-1/3*(8*cos(b*x + a)^4 - 12*cos(b*x + a)^2 + 3)/((b*cos(b*x + a)^3 - b*cos(b*x + a))*sin(b*x + a))

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giac [A]  time = 0.22, size = 35, normalized size = 0.95 \[ -\frac {\frac {6 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2/sin(b*x+a)^4,x, algorithm="giac")

[Out]

-1/3*((6*tan(b*x + a)^2 + 1)/tan(b*x + a)^3 - 3*tan(b*x + a))/b

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maple [A]  time = 0.04, size = 50, normalized size = 1.35 \[ \frac {-\frac {1}{3 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^2/sin(b*x+a)^4,x)

[Out]

1/b*(-1/3/sin(b*x+a)^3/cos(b*x+a)+4/3/sin(b*x+a)/cos(b*x+a)-8/3*cot(b*x+a))

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maxima [A]  time = 0.53, size = 35, normalized size = 0.95 \[ -\frac {\frac {6 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2/sin(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*((6*tan(b*x + a)^2 + 1)/tan(b*x + a)^3 - 3*tan(b*x + a))/b

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mupad [B]  time = 0.42, size = 36, normalized size = 0.97 \[ \frac {\mathrm {tan}\left (a+b\,x\right )}{b}-\frac {2\,{\mathrm {tan}\left (a+b\,x\right )}^2+\frac {1}{3}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(a + b*x)^2*sin(a + b*x)^4),x)

[Out]

tan(a + b*x)/b - (2*tan(a + b*x)^2 + 1/3)/(b*tan(a + b*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**2/sin(b*x+a)**4,x)

[Out]

Integral(sec(a + b*x)**2/sin(a + b*x)**4, x)

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