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

Optimal. Leaf size=23 \[ \frac{\sec (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]

[Out]

-(ArcTanh[Cos[a + b*x]]/b) + Sec[a + b*x]/b

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Rubi [A]  time = 0.0216238, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2622, 321, 207} \[ \frac{\sec (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sec[a + b*x]^2,x]

[Out]

-(ArcTanh[Cos[a + b*x]]/b) + Sec[a + b*x]/b

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0259297, size = 42, normalized size = 1.83 \[ \frac{\sec (a+b x)}{b}+\frac{\log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 34, normalized size = 1.5 \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) }}+{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/cos(b*x+a)+1/b*ln(csc(b*x+a)-cot(b*x+a))

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Maxima [A]  time = 0.986469, size = 49, normalized size = 2.13 \begin{align*} \frac{\frac{2}{\cos \left (b x + a\right )} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.60872, size = 154, normalized size = 6.7 \begin{align*} -\frac{\cos \left (b x + a\right ) \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - \cos \left (b x + a\right ) \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - 2}{2 \, b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (a + b x \right )}}{\sin{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19753, size = 74, normalized size = 3.22 \begin{align*} \frac{\frac{4}{\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1} + \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1) + log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b