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

Optimal. Leaf size=150 \[ \frac{2 (e x)^{m+1} \sqrt{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{-b d n+2 i m+2 i}{4 b d n},-\frac{-5 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n+2 m+2) \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

[Out]

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

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Rubi [A]  time = 0.109999, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4493, 4491, 364} \[ \frac{2 (e x)^{m+1} \sqrt{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac{1}{2},-\frac{2 i m-b d n+2 i}{4 b d n};-\frac{2 i m-5 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n+2 m+2) \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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])

Rubi steps

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

Mathematica [A]  time = 0.517193, size = 131, normalized size = 0.87 \[ -\frac{2 x (e x)^m \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) \text{Hypergeometric2F1}\left (1,-\frac{-3 b d n+2 i m+2 i}{4 b d n},-\frac{-5 b d n+2 i m+2 i}{4 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{(i b d n+2 m+2) \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m}{\frac{1}{\sqrt{\sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{\sqrt{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{\sqrt{\sin{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{\sqrt{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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