3.1.77 \(\int \frac {(e x)^m}{\sin ^{\frac {3}{2}}(d (a+b \log (c x^n)))} \, dx\) [77]

Optimal. Leaf size=150 \[ \frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \, _2F_1\left (\frac {3}{2},-\frac {2 i+2 i m-3 b d n}{4 b d n};-\frac {2 i+2 i m-7 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+3 i b d n) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

[Out]

2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2)*hypergeom([3/2, 1/4*(-2*I-2*I*m+3*b*d*n)/b/d/n],[1/4*(-
2*I-2*I*m+7*b*d*n)/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/e/(2+2*m+3*I*b*d*n)/sin(d*(a+b*ln(c*x^n)))^(3/2)

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Rubi [A]
time = 0.08, 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 = {4581, 4579, 371} \begin {gather*} \frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b d n}\right );-\frac {2 i m-7 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 (3 i b d n+2 m+2) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(3/2),x]

[Out]

(2*(e*x)^(1 + m)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^(3/2)*Hypergeometric2F1[3/2, (3 - ((2*I)*(1 + m))/(b*
d*n))/4, -1/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(2 + 2*m + (3*I)*b*d*n
)*Sin[d*(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 4579

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 4581

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 {3}{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 ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\sin ^{\frac {3}{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 {3}{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 )^{3/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3 i b d}{2}+\frac {1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^{3/2}} \, dx,x,c x^n\right )}{e n \sin ^{\frac {3}{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 )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (1+m)}{b d n}\right );-\frac {2 i+2 i m-7 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+3 i b d n) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(544\) vs. \(2(150)=300\).
time = 5.62, size = 544, normalized size = 3.63 \begin {gather*} \frac {\left (4+8 m+4 m^2+b^2 d^2 n^2\right ) x^{1+i b d n} (e x)^m \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m-3 b d n}{4 b d n};-\frac {2 i+2 i m-7 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+\frac {(-2 i-2 i m+3 b d n) x^{1-i b d n} (e x)^m \left (-2 x^{i b d n} \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} (b d n \cos (b d n \log (x))-2 (1+m) \sin (b d n \log (x)))+(-2 i-2 i m+b d n) \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m+b d n}{4 b d n};-\frac {2 i+2 i m-3 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}\right )}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}}{b d n (-2 i-2 i m+3 b d n) \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left (b d n \cos \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(3/2),x]

[Out]

((4 + 8*m + 4*m^2 + b^2*d^2*n^2)*x^(1 + I*b*d*n)*(e*x)^m*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Hyperge
ometric2F1[1/2, -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a*d)*
(c*x^n)^((2*I)*b*d)] + ((-2*I - (2*I)*m + 3*b*d*n)*x^(1 - I*b*d*n)*(e*x)^m*(-2*x^(I*b*d*n)*Sqrt[((-I)*(-1 + E^
((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[b*d*n*Log[x]] - 2*(1 + m)*Sin[b*d*n*
Log[x]]) + (-2*I - (2*I)*m + b*d*n)*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Hypergeometric2F1[1/2, -1/4*
(2*I + (2*I)*m + b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Sq
rt[Sin[d*(a + b*Log[c*x^n])]]))/Sqrt[Sin[d*(a + b*Log[c*x^n])]])/(b*d*n*(-2*I - (2*I)*m + 3*b*d*n)*Sqrt[((-I)*
(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*
x^n])] + 2*(1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/sin(d*(a+b*ln(c*x^n)))^(3/2),x)

[Out]

int((e*x)^m/sin(d*(a+b*ln(c*x^n)))^(3/2),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((e*x)^m/sin(d*(a+b*log(c*x^n)))^(3/2),x, algorithm="maxima")

[Out]

integrate((x*e)^m/sin((b*log(c*x^n) + a)*d)^(3/2), 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((e*x)^m/sin(d*(a+b*log(c*x^n)))^(3/2),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 \frac {\left (e x\right )^{m}}{\sin ^{\frac {3}{2}}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m/sin(d*(a+b*ln(c*x**n)))**(3/2),x)

[Out]

Integral((e*x)**m/sin(a*d + b*d*log(c*x**n))**(3/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/sin(d*(a+b*log(c*x^n)))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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