3.78 \(\int \frac {(e x)^m}{\sin ^{\frac {5}{2}}(d (a+b \log (c x^n)))} \, dx\)

Optimal. Leaf size=150 \[ \frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \, _2F_1\left (\frac {5}{2},-\frac {2 i m-5 b d n+2 i}{4 b d n};-\frac {2 i m-9 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (5 i b d n+2 m+2) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

[Out]

2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(5/2)*hypergeom([5/2, 1/4*(-2*I-2*I*m+5*b*d*n)/b/d/n],[1/4*(-
2*I-2*I*m+9*b*d*n)/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/e/(2+2*m+5*I*b*d*n)/sin(d*(a+b*ln(c*x^n)))^(5/2)

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Rubi [A]  time = 0.11, antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4493, 4491, 364} \[ \frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (m+1)}{b d n}\right );-\frac {2 i m-9 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (5 i b d n+2 m+2) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*x)^(1 + m)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^(5/2)*Hypergeometric2F1[5/2, (5 - ((2*I)*(1 + m))/(b*
d*n))/4, -(2*I + (2*I)*m - 9*b*d*n)/(4*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(2 + 2*m + (5*I)*b*d*n)*
Sin[d*(a + b*Log[c*x^n])]^(5/2))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4491

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

Rule 4493

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

Rubi steps

\begin {align*} \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\sin ^{\frac {5}{2}}(d (a+b \log (x)))} \, dx,x,c x^n\right )}{e n}\\ &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {5}{2} i b d-\frac {1+m}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {5 i b d}{2}+\frac {1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^{5/2}} \, dx,x,c x^n\right )}{e n \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\\ &=\frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (1+m)}{b d n}\right );-\frac {2 i+2 i m-9 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+5 i b d n) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\\ \end {align*}

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Mathematica [A]  time = 2.44, size = 214, normalized size = 1.43 \[ \frac {2 x (e x)^m \left (-(-i b d n+2 m+2) \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,-\frac {2 i m-3 b d n+2 i}{4 b d n};-\frac {2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-b d n \cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-2 m-2\right )}{3 b^2 d^2 n^2 \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{m}}{\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((e*x)**m/sin(d*(a+b*ln(c*x**n)))**(5/2),x)

[Out]

Timed out

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