3.3.50 \(\int \sec ^3(a+b \log (c x^n)) \, dx\) [250]
Optimal. Leaf size=85 \[ \frac {8 e^{3 i a} x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right );\frac {1}{2} \left (5-\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+3 i b n} \]
[Out]
8*exp(3*I*a)*x*(c*x^n)^(3*I*b)*hypergeom([3, 3/2-1/2*I/b/n],[5/2-1/2*I/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+3*
I*b*n)
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4599, 4601,
371} \begin {gather*} \frac {8 e^{3 i a} x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right );\frac {1}{2} \left (5-\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+3 i b n} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sec[a + b*Log[c*x^n]]^3,x]
[Out]
(8*E^((3*I)*a)*x*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 - I/(b*n))/2, (5 - I/(b*n))/2, -(E^((2*I)*a)*(c*x^n
)^((2*I)*b))])/(1 + (3*I)*b*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 4599
Int[Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])
Rule 4601
Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[2^p*E^(I*a*d*p), Int[(e*x)^
m*(x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
Rubi steps
\begin {align*} \int \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \sec ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (8 e^{3 i a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+3 i b+\frac {1}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=\frac {8 e^{3 i a} x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right );\frac {1}{2} \left (5-\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+3 i b n}\\ \end {align*}
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Mathematica [A]
time = 5.48, size = 120, normalized size = 1.41 \begin {gather*} \frac {x \left (2 e^{i a} (1-i b n) \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {i}{2 b n};\frac {3}{2}-\frac {i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec \left (a+b \log \left (c x^n\right )\right ) \left (-1+b n \tan \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sec[a + b*Log[c*x^n]]^3,x]
[Out]
(x*(2*E^(I*a)*(1 - I*b*n)*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), -E^((2*I)*(
a + b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]*(-1 + b*n*Tan[a + b*Log[c*x^n]])))/(2*b^2*n^2)
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \sec ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(a+b*ln(c*x^n))^3,x)
[Out]
int(sec(a+b*ln(c*x^n))^3,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(sec(a+b*log(c*x^n))^3,x, algorithm="maxima")
[Out]
-((b*n*sin(b*log(c)) + cos(b*log(c)))*x*cos(b*log(x^n) + a) + (b*n*cos(b*log(c)) - sin(b*log(c)))*x*sin(b*log(
x^n) + a) + (((b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(3*b*log(c)))*n + cos(4*b*log(c))*cos(
3*b*log(c)) + sin(4*b*log(c))*sin(3*b*log(c)))*x*cos(3*b*log(x^n) + 3*a) - ((b*cos(b*log(c))*sin(4*b*log(c)) -
b*cos(4*b*log(c))*sin(b*log(c)))*n - cos(4*b*log(c))*cos(b*log(c)) - sin(4*b*log(c))*sin(b*log(c)))*x*cos(b*l
og(x^n) + a) - ((b*cos(4*b*log(c))*cos(3*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)))*n - cos(3*b*log(c))*si
n(4*b*log(c)) + cos(4*b*log(c))*sin(3*b*log(c)))*x*sin(3*b*log(x^n) + 3*a) + ((b*cos(4*b*log(c))*cos(b*log(c))
+ b*sin(4*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*x*sin(b
*log(x^n) + a))*cos(4*b*log(x^n) + 4*a) - (2*((b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2*b*l
og(c)))*n - cos(3*b*log(c))*cos(2*b*log(c)) - sin(3*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - 2*(
(b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(3*b*log(c)) -
cos(3*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) + (b*n*sin(3*b*log(c)) - cos(3*b*log(c)))*x)*cos(3*
b*log(x^n) + 3*a) - 2*(((b*cos(b*log(c))*sin(2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)))*n - cos(2*b*log(c)
)*cos(b*log(c)) - sin(2*b*log(c))*sin(b*log(c)))*x*cos(b*log(x^n) + a) - ((b*cos(2*b*log(c))*cos(b*log(c)) + b
*sin(2*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)))*x*sin(b*log
(x^n) + a))*cos(2*b*log(x^n) + 2*a) - (b^4*n^4*cos(b*log(c)) + b^2*n^2*cos(b*log(c)) + ((b^4*cos(4*b*log(c))^2
*cos(b*log(c)) + b^4*cos(b*log(c))*sin(4*b*log(c))^2)*n^4 + (b^2*cos(4*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*l
og(c))*sin(4*b*log(c))^2)*n^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^4*cos(2*b*log(c))^2*cos(b*log(c)) + b^4*cos(b
*log(c))*sin(2*b*log(c))^2)*n^4 + (b^2*cos(2*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(2*b*log(c))^2)*
n^2)*cos(2*b*log(x^n) + 2*a)^2 + ((b^4*cos(4*b*log(c))^2*cos(b*log(c)) + b^4*cos(b*log(c))*sin(4*b*log(c))^2)*
n^4 + (b^2*cos(4*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(4*b*log(c))^2)*n^2)*sin(4*b*log(x^n) + 4*a)
^2 + 4*((b^4*cos(2*b*log(c))^2*cos(b*log(c)) + b^4*cos(b*log(c))*sin(2*b*log(c))^2)*n^4 + (b^2*cos(2*b*log(c))
^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(2*b*log(c))^2)*n^2)*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^4*n^4*cos(4*b*lo
g(c))*cos(b*log(c)) + b^2*n^2*cos(4*b*log(c))*cos(b*log(c)) + 2*((b^4*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*lo
g(c)) + b^4*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))*n^4 + (b^2*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*lo
g(c)) + b^2*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))*n^2)*cos(2*b*log(x^n) + 2*a) + 2*((b^4*cos(2*b*log(
c))*cos(b*log(c))*sin(4*b*log(c)) - b^4*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^4 + (b^2*cos(2*b*log(
c))*cos(b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^2)*sin(2*b*log(x^n) +
2*a))*cos(4*b*log(x^n) + 4*a) + 4*(b^4*n^4*cos(2*b*log(c))*cos(b*log(c)) + b^2*n^2*cos(2*b*log(c))*cos(b*log(
c)))*cos(2*b*log(x^n) + 2*a) - 2*(b^4*n^4*cos(b*log(c))*sin(4*b*log(c)) + b^2*n^2*cos(b*log(c))*sin(4*b*log(c)
) + 2*((b^4*cos(2*b*log(c))*cos(b*log(c))*sin(4*b*log(c)) - b^4*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))
*n^4 + (b^2*cos(2*b*log(c))*cos(b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))
*n^2)*cos(2*b*log(x^n) + 2*a) - 2*((b^4*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^4*cos(b*log(c))*sin(
4*b*log(c))*sin(2*b*log(c)))*n^4 + (b^2*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^2*cos(b*log(c))*sin(
4*b*log(c))*sin(2*b*log(c)))*n^2)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) - 4*(b^4*n^4*cos(b*log(c))*
sin(2*b*log(c)) + b^2*n^2*cos(b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(((cos(2*b*log(c))*
cos(b*log(x^n) + a) + sin(2*b*log(c))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) - (cos(b*log(x^n) + a)*sin(
2*b*log(c)) - cos(2*b*log(c))*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) + cos(b*log(x^n) + a))/(2*b^2*n^2*c
os(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^2*n^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b^2*cos(2*b*log(
c))^2 + b^2*sin(2*b*log(c))^2)*n^2*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)
*n^2*sin(2*b*log(x^n) + 2*a)^2 + b^2*n^2), x) - (b^4*n^4*sin(b*log(c)) + b^2*n^2*sin(b*log(c)) + ((b^4*cos(4*b
*log(c))^2*sin(b*log(c)) + b^4*sin(4*b*log(c))^2*sin(b*log(c)))*n^4 + (b^2*cos(4*b*log(c))^2*sin(b*log(c)) + b
^2*sin(4*b*log(c))^2*sin(b*log(c)))*n^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^4*cos(2*b*log(c))^2*sin(b*log(c)) +
b^4*sin(2*b*log(c))^2*sin(b*log(c)))*n^4 + (b^2*cos(2*b*log(c))^2*sin(b*log(c)) + b^2*sin(2*b*log(c))^2*sin(b
*log(c)))*n^2)*cos(2*b*log(x^n) + 2*a)^2 + ((b^4*cos(4*b*log(c))^2*sin(b*log(c)) + b^4*sin(4*b*log(c))^2*sin(b
*log(c)))*n^4 + (b^2*cos(4*b*log(c))^2*sin(b*log(c)) + b^2*sin(4*b*log(c))^2*sin(b*log(c)))*n^2)*sin(4*b*log(x
^n) + 4*a)^2 + 4*((b^4*cos(2*b*log(c))^2*sin(b*...
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*log(c*x^n))^3,x, algorithm="fricas")
[Out]
integral(sec(b*log(c*x^n) + a)^3, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sec ^{3}{\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(sec(a+b*ln(c*x**n))**3,x)
[Out]
Integral(sec(a + b*log(c*x**n))**3, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*log(c*x^n))^3,x, algorithm="giac")
[Out]
integrate(sec(b*log(c*x^n) + a)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/cos(a + b*log(c*x^n))^3,x)
[Out]
int(1/cos(a + b*log(c*x^n))^3, x)
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