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*hypergeom([1/2, -1/2*m],[1-1/2*m],1/c^2/x^2)/c/m/(1+m)

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Rubi [A]  time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.19, 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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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m}\,{d x} \]

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)

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maple [F]  time = 3.87, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d^{m} x x^{m} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - {\left (c^{2} d^{m} m + c^{2} d^{m}\right )} \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} x^{m}}{c^{2} m - {\left (c^{4} m + c^{4}\right )} x^{2} + c^{2}}\,{d x}\right )} b}{m + 1} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^m*x*x^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^2*d^m*m + c^2*d^m)*integrate(-sqrt(c*x + 1)*sqrt(c*x -
1)*x^m/(c^2*m - (c^4*m + c^4)*x^2 + c^2), x))*b/(m + 1) + (d*x)^(m + 1)*a/(d*(m + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*asin(1/(c*x))),x)

[Out]

int((d*x)^m*(a + b*asin(1/(c*x))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )\, dx \]

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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