3.41 \(\int (d x)^m (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=66 \[ \frac{(d x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{d (m+1)}+\frac{b (d x)^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{c^2 x^2}\right )}{c m (m+1)} \]

[Out]

((d*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(d*(1 + m)) + (b*(d*x)^m*Hypergeometric2F1[1/2, -m/2, 1 - m/2, 1/(c^2*x^2)
])/(c*m*(1 + m))

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Rubi [A]  time = 0.043359, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5221, 339, 364} \[ \frac{(d x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{d (m+1)}+\frac{b (d x)^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{c^2 x^2}\right )}{c m (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcCsc[c*x]),x]

[Out]

((d*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(d*(1 + m)) + (b*(d*x)^m*Hypergeometric2F1[1/2, -m/2, 1 - m/2, 1/(c^2*x^2)
])/(c*m*(1 + m))

Rule 5221

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsc[c*x]
))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c
, d, m}, x] && NeQ[m, -1]

Rule 339

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst
[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac{(b d) \int \frac{(d x)^{-1+m}}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{c (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}-\frac{\left (b \left (\frac{1}{x}\right )^m (d x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac{b (d x)^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{c^2 x^2}\right )}{c m (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.172998, size = 83, normalized size = 1.26 \[ \frac{(d x)^m \left ((m+1) x \left (a+b \csc ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right )}{c \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{(m+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*ArcCsc[c*x]),x]

[Out]

((d*x)^m*((1 + m)*x*(a + b*ArcCsc[c*x]) + (b*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^
2*x^2])/(c*Sqrt[1 - 1/(c^2*x^2)])))/(1 + m)^2

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Maple [F]  time = 2.089, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*arccsc(c*x)),x)

[Out]

int((d*x)^m*(a+b*arccsc(c*x)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccsc(c*x) + a)*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*acsc(c*x)),x)

[Out]

Integral((d*x)**m*(a + b*acsc(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)*(d*x)^m, x)