3.2.30 \(\int \frac {x^m}{\cos ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [130]

Optimal. Leaf size=130 \[ \frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},-\frac {2 i+2 i m-3 b n}{4 b n};-\frac {2 i+2 i m-7 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4582, 4580, 371} \begin {gather*} \frac {2 x^{m+1} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b n}\right );-\frac {2 i m-7 b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(3 i b n+2 m+2) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1 + m)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2)*Hypergeometric2F1[3/2, (3 - ((2*I)*(1 + m))/(b*n))/4, -
1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + 2*m + (3*I)*b*n)*Cos[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 4580

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[Cos[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 4582

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cos[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 {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\cos ^{\frac {3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {3 i b}{2}-\frac {1+m}{n}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3 i b}{2}+\frac {1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^{3/2}} \, dx,x,c x^n\right )}{n \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (1+m)}{b n}\right );-\frac {2 i+2 i m-7 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\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. \(487\) vs. \(2(130)=260\).
time = 5.70, size = 487, normalized size = 3.75 \begin {gather*} -\frac {x^{1+m-i b n} \left (\left (4+8 m+4 m^2+b^2 n^2\right ) x^{2 i b n} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m-3 b n}{4 b n};-\frac {2 i+2 i m-7 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i-2 i m+3 b n) \left ((-2 i-2 i m+b n) \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n};-\frac {2 i+2 i m-3 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )-2 x^{i b n} \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} (b n \cos (b n \log (x))-2 (1+m) \sin (b n \log (x)))\right )\right )}{b n (-2 i-2 i m+3 b n) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (-2 (1+m) \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+b n \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m/cos(a + b*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(x^m/cos(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 {x^m}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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