3.16 \(\int \csc (c+d x) (a+b \tan (c+d x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.029023, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3517, 3770} \[ \frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3517

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Expand[Sin[e
+ f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]

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

Mathematica [A]  time = 0.0237081, size = 52, normalized size = 2. \[ \frac{a \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.036, size = 42, normalized size = 1.6 \begin{align*}{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{b\ln \left ( \sec \left ( dx+c \right ) +\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*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.53495, size = 62, normalized size = 2.38 \begin{align*} \frac{b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, a \log \left (\cot \left (d x + c\right ) + \csc \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*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/2*(b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 2*a*log(cot(d*x + c) + csc(d*x + c)))/d

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Fricas [B]  time = 2.35074, size = 170, normalized size = 6.54 \begin{align*} -\frac{a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.41334, size = 66, normalized size = 2.54 \begin{align*} \frac{b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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