3.5 \(\int x (a+b \csc (c+d x^2)) \, dx\)

Optimal. Leaf size=26 \[ \frac{a x^2}{2}-\frac{b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0233589, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {14, 4205, 3770} \[ \frac{a x^2}{2}-\frac{b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Csc[c + d*x^2]),x]

[Out]

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

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]]

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

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

Mathematica [B]  time = 0.0266236, size = 57, normalized size = 2.19 \[ \frac{a x^2}{2}+\frac{b \log \left (\sin \left (\frac{c}{2}+\frac{d x^2}{2}\right )\right )}{2 d}-\frac{b \log \left (\cos \left (\frac{c}{2}+\frac{d x^2}{2}\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Csc[c + d*x^2]),x]

[Out]

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

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Maple [A]  time = 0.019, size = 39, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}}{2}}-{\frac{b\ln \left ( \csc \left ( d{x}^{2}+c \right ) +\cot \left ( d{x}^{2}+c \right ) \right ) }{2\,d}}+{\frac{ac}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*csc(d*x^2+c)),x)

[Out]

1/2*a*x^2-1/2/d*b*ln(csc(d*x^2+c)+cot(d*x^2+c))+1/2/d*a*c

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Maxima [A]  time = 0.968429, size = 42, normalized size = 1.62 \begin{align*} \frac{1}{2} \, a x^{2} - \frac{b \log \left (\cot \left (d x^{2} + c\right ) + \csc \left (d x^{2} + c\right )\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*csc(d*x^2+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.510691, size = 119, normalized size = 4.58 \begin{align*} \frac{2 \, a d x^{2} - b \log \left (\frac{1}{2} \, \cos \left (d x^{2} + c\right ) + \frac{1}{2}\right ) + b \log \left (-\frac{1}{2} \, \cos \left (d x^{2} + c\right ) + \frac{1}{2}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*csc(d*x^2+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.43082, size = 42, normalized size = 1.62 \begin{align*} \begin{cases} \frac{a \left (c + d x^{2}\right ) - b \log{\left (\cot{\left (c + d x^{2} \right )} + \csc{\left (c + d x^{2} \right )} \right )}}{2 d} & \text{for}\: d \neq 0 \\\frac{x^{2} \left (a + b \csc{\left (c \right )}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*csc(d*x**2+c)),x)

[Out]

Piecewise(((a*(c + d*x**2) - b*log(cot(c + d*x**2) + csc(c + d*x**2)))/(2*d), Ne(d, 0)), (x**2*(a + b*csc(c))/
2, True))

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Giac [A]  time = 1.21211, size = 41, normalized size = 1.58 \begin{align*} \frac{{\left (d x^{2} + c\right )} a + b \log \left ({\left | \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right ) \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*csc(d*x^2+c)),x, algorithm="giac")

[Out]

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