3.152 \(\int \csc ^3(a+b x) \sec (a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{\log (\tan (a+b x))}{b}-\frac{\cot ^2(a+b x)}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0218528, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2620, 14} \[ \frac{\log (\tan (a+b x))}{b}-\frac{\cot ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sec[a + b*x],x]

[Out]

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

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]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.041896, size = 34, normalized size = 1.26 \[ -\frac{\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sec[a + b*x],x]

[Out]

-(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]])/(2*b)

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Maple [A]  time = 0.018, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}b}}+{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)/sin(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.99941, size = 49, normalized size = 1.81 \begin{align*} -\frac{\frac{1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)/sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b

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Fricas [B]  time = 2.38227, size = 176, normalized size = 6.52 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) -{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right ) - 1}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)/sin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*((cos(b*x + a)^2 - 1)*log(cos(b*x + a)^2) - (cos(b*x + a)^2 - 1)*log(-1/4*cos(b*x + a)^2 + 1/4) - 1)/(b*c
os(b*x + a)^2 - b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (a + b x \right )}}{\sin ^{3}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)/sin(b*x+a)**3,x)

[Out]

Integral(sec(a + b*x)/sin(a + b*x)**3, x)

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Giac [B]  time = 1.16666, size = 161, normalized size = 5.96 \begin{align*} -\frac{\frac{{\left (\frac{4 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 4 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 8 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)/sin(b*x+a)^3,x, algorithm="giac")

[Out]

-1/8*((4*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) - (cos(b*x + a) - 1)
/(cos(b*x + a) + 1) - 4*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) + 8*log(abs(-(cos(b*x + a) - 1)/(cos
(b*x + a) + 1) - 1)))/b