3.3.91 \(\int \csc (a+b \log (c x^n)) \, dx\) [291]

Optimal. Leaf size=84 \[ \frac {2 e^{i a} x \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right );\frac {1}{2} \left (3-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-b n} \]

[Out]

2*exp(I*a)*x*(c*x^n)^(I*b)*hypergeom([1, 1/2-1/2*I/b/n],[3/2-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/(I-b*n)

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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4600, 4602, 371} \begin {gather*} \frac {2 e^{i a} x \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right );\frac {1}{2} \left (3-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+i} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]],x]

[Out]

(2*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1 - I/(b*n))/2, (3 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)^((2*I)*b
)])/(I - b*n)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4600

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4602

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(-2*I)^p*E^(I*a*d*p), Int[(
e*x)^m*(x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \csc \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=-\frac {\left (2 i e^{i a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {1}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 e^{i a} x \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right );\frac {1}{2} \left (3-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-b n}\\ \end {align*}

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Mathematica [A]
time = 1.28, size = 80, normalized size = 0.95 \begin {gather*} -\frac {2 e^{i a} x \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {i}{2 b n};\frac {3}{2}-\frac {i}{2 b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-i+b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*Log[c*x^n]],x]

[Out]

(-2*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x
^n]))])/(-I + b*n)

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \csc \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(a+b*ln(c*x^n)),x)

[Out]

int(csc(a+b*ln(c*x^n)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*log(c*x^n) + a), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csc(b*log(c*x^n) + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*ln(c*x**n)),x)

[Out]

Integral(csc(a + b*log(c*x**n)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*log(c*x^n) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*log(c*x^n)),x)

[Out]

int(1/sin(a + b*log(c*x^n)), x)

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