3.68 \(\int \frac {1}{x \sin ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=68 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

-2/3*(sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))*EllipticF(cos(1/2*a+1/4*Pi+
1/2*b*ln(c*x^n)),2^(1/2))/b/n-2/3*cos(a+b*ln(c*x^n))/b/n/sin(a+b*ln(c*x^n))^(3/2)

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Rubi [A]  time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2636, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sin[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(2*EllipticF[(a - Pi/2 + b*Log[c*x^n])/2, 2])/(3*b*n) - (2*Cos[a + b*Log[c*x^n]])/(3*b*n*Sin[a + b*Log[c*x^n]]
^(3/2))

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sin ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 61, normalized size = 0.90 \[ \frac {2 \left (F\left (\left .\frac {1}{4} \left (2 a+2 b \log \left (c x^n\right )-\pi \right )\right |2\right )-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sin[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(2*(EllipticF[(2*a - Pi + 2*b*Log[c*x^n])/4, 2] - Cos[a + b*Log[c*x^n]]/Sin[a + b*Log[c*x^n]]^(3/2)))/(3*b*n)

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{2} - x\right )} \sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

integral(-1/((x*cos(b*log(c*x^n) + a)^2 - x)*sqrt(sin(b*log(c*x^n) + a))), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

integrate(1/(x*sin(b*log(c*x^n) + a)^(5/2)), x)

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maple [A]  time = 0.08, size = 131, normalized size = 1.93 \[ \frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 n \sin \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/sin(a+b*ln(c*x^n))^(5/2),x)

[Out]

1/3/n/sin(a+b*ln(c*x^n))^(3/2)*((sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^
n)))^(1/2)*EllipticF((sin(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))*sin(a+b*ln(c*x^n))-2*cos(a+b*ln(c*x^n))^2)/cos(
a+b*ln(c*x^n))/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*sin(b*log(c*x^n) + a)^(5/2)), x)

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mupad [B]  time = 2.96, size = 65, normalized size = 0.96 \[ -\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*sin(a + b*log(c*x^n))^(5/2)),x)

[Out]

-(cos(a + b*log(c*x^n))*(sin(a + b*log(c*x^n))^2)^(3/4)*hypergeom([1/2, 7/4], 3/2, cos(a + b*log(c*x^n))^2))/(
b*n*sin(a + b*log(c*x^n))^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

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