3.164 \(\int \csc (c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=26 \[ \frac{b \log (\tan (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d} \]

[Out]

-((a*ArcTanh[Cos[c + d*x]])/d) + (b*Log[Tan[c + d*x]])/d

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Rubi [A]  time = 0.0725366, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3872, 2834, 2620, 29, 3770} \[ \frac{b \log (\tan (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]*(a + b*Sec[c + d*x]),x]

[Out]

-((a*ArcTanh[Cos[c + d*x]])/d) + (b*Log[Tan[c + d*x]])/d

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [B]  time = 0.0347635, size = 63, normalized size = 2.42 \[ \frac{a \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.033, size = 35, normalized size = 1.4 \begin{align*}{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)*(a+b*sec(d*x+c)),x)

[Out]

1/d*a*ln(csc(d*x+c)-cot(d*x+c))+b*ln(tan(d*x+c))/d

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Maxima [A]  time = 0.953032, size = 61, normalized size = 2.35 \begin{align*} -\frac{{\left (a - b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) -{\left (a + b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 2 \, b \log \left (\cos \left (d x + c\right )\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72636, size = 149, normalized size = 5.73 \begin{align*} -\frac{2 \, b \log \left (-\cos \left (d x + c\right )\right ) +{\left (a - b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a + b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a+b*sec(d*x+c)),x)

[Out]

Integral((a + b*sec(c + d*x))*csc(c + d*x), x)

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Giac [B]  time = 1.34591, size = 82, normalized size = 3.15 \begin{align*} \frac{{\left (a + b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

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