\(\int \frac {(e x)^m}{\sqrt {\sin (d (a+b \log (c x^n)))}} \, dx\) [76]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\frac {2 (e x)^{1+m} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m-b d n}{4 b d n},-\frac {2 i+2 i m-5 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+i b d n) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4581, 4579, 371} \[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\frac {2 (e x)^{m+1} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i m-b d n+2 i}{4 b d n},-\frac {2 i m-5 b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n+2 m+2) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

[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, -1/4*(2*I + (2*I)*m - b*d*
n)/(b*d*n), -1/4*(2*I + (2*I)*m - 5*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(2 + 2*m + I*b*d*n)
*Sqrt[Sin[d*(a + b*Log[c*x^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 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*} \text {integral}& = \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}}}{\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 ) \text {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}} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.76 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.10 \[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\frac {2 \sqrt {2-2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )}} x (e x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {-2 i-2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-5 b d n}{4 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {-i e^{-i d \left (a+b \log \left (c x^n\right )\right )} \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )} (2+2 m+i b d n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (e x \right )^{m}}{\sqrt {\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((e*x)^m/sin(d*(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)

Sympy [F]

\[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\int \frac {\left (e x\right )^{m}}{\sqrt {\sin {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\int { \frac {\left (e x\right )^{m}}{\sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\int { \frac {\left (e x\right )^{m}}{\sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx=\int \frac {{\left (e\,x\right )}^m}{\sqrt {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}} \,d x \]

[In]

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

[Out]

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