\(\int \frac {1}{\csc ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [316]

Optimal result

Integrand size = 15, antiderivative size = 109 \[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2*x*hypergeom([-3/2, -3/4-1/2*I/b/n],[1/4-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/(2-3*I*b*n)/(1-exp(2*I*a)*(c*
x^n)^(2*I*b))^(3/2)/csc(a+b*ln(c*x^n))^(3/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4600, 4604, 371} \[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[Csc[a + b*Log[c*x^n]]^(-3/2),x]

[Out]

(2*x*Hypergeometric2F1[-3/2, (-3 - (2*I)/(b*n))/4, (1 - (2*I)/(b*n))/4, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((2 -
(3*I)*b*n)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2))

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 4604

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1}{n}}}{\csc ^{\frac {3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1}{n}}\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b}{2}+\frac {1}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ & = \frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 i x \left ((2-i b n) \left (-2+3 b n \cot \left (a+b \log \left (c x^n\right )\right )\right )-3 b^2 e^{-2 i a} n^2 \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {i}{2 b n},\frac {5}{4}+\frac {i}{2 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{(2 i-3 b n) (2 i+b n) (2 i+3 b n) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[Csc[a + b*Log[c*x^n]]^(-3/2),x]

[Out]

((2*I)*x*((2 - I*b*n)*(-2 + 3*b*n*Cot[a + b*Log[c*x^n]]) - (3*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Csc
[a + b*Log[c*x^n]]^2*Hypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((-2*I)*(a + b*Log[c*x^n]))]
)/(E^((2*I)*a)*(c*x^n)^((2*I)*b))))/((2*I - 3*b*n)*(2*I + b*n)*(2*I + 3*b*n)*Csc[a + b*Log[c*x^n]]^(3/2))

Maple [F]

\[\int \frac {1}{{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}d x\]

[In]

int(1/csc(a+b*ln(c*x^n))^(3/2),x)

[Out]

int(1/csc(a+b*ln(c*x^n))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/csc(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\csc ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/csc(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(csc(a + b*log(c*x**n))**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/csc(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(csc(b*log(c*x^n) + a)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/csc(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(csc(b*log(c*x^n) + a)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/(1/sin(a + b*log(c*x^n)))^(3/2),x)

[Out]

int(1/(1/sin(a + b*log(c*x^n)))^(3/2), x)