\(\int \frac {1}{\sqrt {\csc (a+b \log (c x^n))}} \, dx\) [314]

Optimal result

Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 110, 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} \[ \int \frac {1}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {b n+2 i}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/Sqrt[Csc[a + b*Log[c*x^n]]],x]

[Out]

(2*x*Hypergeometric2F1[-1/2, -1/4*(2*I + b*n)/(b*n), (3 - (2*I)/(b*n))/4, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((2
- I*b*n)*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Csc[a + b*Log[c*x^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 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}}}{\sqrt {\csc (a+b \log (x))}} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{\frac {i b}{2}-\frac {1}{n}}\right ) \text {Subst}\left (\int x^{-1-\frac {i b}{2}+\frac {1}{n}} \sqrt {1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \\ & = \frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(110)=220\).

Time = 3.21 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.43 \[ \int \frac {1}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 b e^{i a} n x \left (c x^n\right )^{i b} \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((2 i+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(2 i+b n) (-2 i+3 b n) \left ((2 i+b n) x^{2 i b n}+e^{2 i a} (-2 i+b n) \left (c x^n\right )^{2 i b}\right )}+\frac {2 x \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (b n \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+2 \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \]

[In]

Integrate[1/Sqrt[Csc[a + b*Log[c*x^n]]],x]

[Out]

(-2*b*E^(I*a)*n*x*(c*x^n)^(I*b)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 +
E^((2*I)*a)*(c*x^n)^((2*I)*b))]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I
/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4
- (I/2)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)]))/((2*I + b*n)*(-2*I + 3*b*n)*((2*I + b*n)*x^((2*I)*b*n) + E^((2
*I)*a)*(-2*I + b*n)*(c*x^n)^((2*I)*b))) + (2*x*Sin[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Csc[a + b*Log[c*x^n]]
]*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(csc(a + b*log(c*x**n))), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(csc(b*log(c*x^n) + a)), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/sqrt(csc(b*log(c*x^n) + a)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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