3.4.8 \(\int \sqrt {\csc (a+b \log (c x^n))} \, dx\) [308]

Optimal. Leaf size=109 \[ \frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n} \]

[Out]

2*x*hypergeom([1/2, 1/4-1/2*I/b/n],[5/4-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^
(1/2)*csc(a+b*ln(c*x^n))^(1/2)/(2+I*b*n)

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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4600, 4604, 371} \begin {gather*} \frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csc[a + b*Log[c*x^n]]],x]

[Out]

(2*x*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Csc[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, (1 - (2*I)/(b*
n))/4, (5 - (2*I)/(b*n))/4, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(2 + 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 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*} \int \sqrt {\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}} \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}} \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 )}\right ) \text {Subst}\left (\int \frac {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}\\ &=\frac {2 x \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 )} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 115, normalized size = 1.06 \begin {gather*} \frac {2 i e^{-2 i a} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) x \left (c x^n\right )^{-2 i b} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\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 )}{2 i+b n} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Sqrt[Csc[a + b*Log[c*x^n]]]*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)*(2*I + b*n)*(c*x^n)^((2*I)*b))

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \sqrt {\csc }\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\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))**(1/2),x)

[Out]

Integral(sqrt(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))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(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 \sqrt {\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)))^(1/2),x)

[Out]

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

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