3.3.54 \(\int x \sec ^4(a+b \log (c x^n)) \, dx\) [254]

Optimal. Leaf size=79 \[ \frac {8 e^{4 i a} x^2 \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2-\frac {i}{b n};3-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4601, 371} \begin {gather*} \frac {8 e^{4 i a} x^2 \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2-\frac {i}{b n};3-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sec[a + b*Log[c*x^n]]^4,x]

[Out]

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

Rule 4605

Int[((e_.)*(x_))^(m_.)*Sec[((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)*Sec[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 x \sec ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sec ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (16 e^{4 i a} x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+4 i b+\frac {2}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac {8 e^{4 i a} x^2 \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2-\frac {i}{b n};3-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n}\\ \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. \(204\) vs. \(2(79)=158\).
time = 13.52, size = 204, normalized size = 2.58 \begin {gather*} \frac {x^2 \left (2 e^{2 i a} (i+b n) \left (c x^n\right )^{2 i b} \, _2F_1\left (1,1-\frac {i}{b n};2-\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-2 i \left (1+b^2 n^2\right ) \, _2F_1\left (1,-\frac {i}{b n};1-\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec ^2\left (a+b \log \left (c x^n\right )\right ) \left (-b n+\left (1+2 b^2 n^2+\left (1+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b^3 n^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sec[a + b*Log[c*x^n]]^4,x]

[Out]

(x^2*(2*E^((2*I)*a)*(I + b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - I/(b*n), 2 - I/(b*n), -E^((2*I)*(a +
b*Log[c*x^n]))] - (2*I)*(1 + b^2*n^2)*Hypergeometric2F1[1, (-I)/(b*n), 1 - I/(b*n), -E^((2*I)*(a + b*Log[c*x^n
]))] + Sec[a + b*Log[c*x^n]]^2*(-(b*n) + (1 + 2*b^2*n^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Tan[a + b*L
og[c*x^n]])))/(3*b^3*n^3)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x \left (\sec ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sec(a+b*ln(c*x^n))^4,x)

[Out]

int(x*sec(a+b*ln(c*x^n))^4,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*sec(a+b*log(c*x^n))^4,x, algorithm="maxima")

[Out]

-4/3*(3*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x^2*cos(4*b*log(x^n) + 4*a)^2 + 3*(b*cos(2*b*log(c))^2 +
b*sin(2*b*log(c))^2)*n*x^2*cos(2*b*log(x^n) + 2*a)^2 + 3*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x^2*si
n(4*b*log(x^n) + 4*a)^2 + 3*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*x^2*sin(2*b*log(x^n) + 2*a)^2 + (b*n
*cos(2*b*log(c)) - sin(2*b*log(c)))*x^2*cos(2*b*log(x^n) + 2*a) - (b*n*sin(2*b*log(c)) + cos(2*b*log(c)))*x^2*
sin(2*b*log(x^n) + 2*a) + (((b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n - cos(4*
b*log(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x^2*cos(4*b*log(x^n) + 4*a) - (3*(b^2*cos(2*b*log
(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 - (b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6
*b*log(c))*sin(2*b*log(c)))*n + 2*cos(2*b*log(c))*sin(6*b*log(c)) - 2*cos(6*b*log(c))*sin(2*b*log(c)))*x^2*cos
(2*b*log(x^n) + 2*a) + ((b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n + cos(6*b*lo
g(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x^2*sin(4*b*log(x^n) + 4*a) + (3*(b^2*cos(6*b*log(c))
*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*l
og(c))*sin(2*b*log(c)))*n + 2*cos(6*b*log(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(c)))*x^2*sin(2*b
*log(x^n) + 2*a) - (b^2*n^2*sin(6*b*log(c)) + sin(6*b*log(c)))*x^2)*cos(6*b*log(x^n) + 6*a) - (3*(3*(b^2*cos(2
*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 - 2*(b*cos(4*b*log(c))*cos(2*b*log(c)) +
b*sin(4*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*x^2
*cos(2*b*log(x^n) + 2*a) - 3*(3*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^
2 + 2*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(
c)) + sin(4*b*log(c))*sin(2*b*log(c)))*x^2*sin(2*b*log(x^n) + 2*a) + (3*b^2*n^2*sin(4*b*log(c)) - b*n*cos(4*b*
log(c)) + 2*sin(4*b*log(c)))*x^2)*cos(4*b*log(x^n) + 4*a) + 18*(b^8*n^8 + b^6*n^6 + ((b^8*cos(6*b*log(c))^2 +
b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*cos(6*b*log(x^n) + 6*a)^2 +
9*((b^8*cos(4*b*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*
cos(4*b*log(x^n) + 4*a)^2 + 9*((b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 +
b^6*sin(2*b*log(c))^2)*n^6)*cos(2*b*log(x^n) + 2*a)^2 + ((b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 +
(b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*sin(6*b*log(x^n) + 6*a)^2 + 9*((b^8*cos(4*b*log(c))^2 +
b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*sin(4*b*log(x^n) + 4*a)^2 +
9*((b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*
sin(2*b*log(x^n) + 2*a)^2 + 2*(b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)) + 3*((b^8*cos(6*b*log(c))*cos
(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*l
og(c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a) + 3*((b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*l
og(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)
*cos(2*b*log(x^n) + 2*a) + 3*((b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8
+ (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) + 3
*((b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6
*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b
^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)) + 3*((b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c
))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos
(2*b*log(x^n) + 2*a) + 3*((b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b
^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(4*
b*log(x^n) + 4*a) + 6*(b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(b^8*n^8
*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)) + 3*((b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin
(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*cos(4*b*l
og(x^n) + 4*a) + 3*((b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos
(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) - 3*((b^8*cos
(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c))
+ b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) - 3*((b^8*cos(6*b*log(c))*cos(2*b*log(c))
+ b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (...

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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(x*sec(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

integral(x*sec(b*log(c*x^n) + a)^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sec ^{4}{\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*sec(a+b*ln(c*x**n))**4,x)

[Out]

Integral(x*sec(a + b*log(c*x**n))**4, 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(x*sec(a+b*log(c*x^n))^4,x, algorithm="giac")

[Out]

integrate(x*sec(b*log(c*x^n) + a)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cos(a + b*log(c*x^n))^4,x)

[Out]

int(x/cos(a + b*log(c*x^n))^4, x)

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