3.2.77 \(\int (e x)^m \tan ^3(d (a+b \log (c x^n))) \, dx\) [177]

Optimal. Leaf size=351 \[ -\frac {(i (1+m)-b d n) (1+m+2 i b d n) (e x)^{1+m}}{2 b^2 d^2 e (1+m) n^2}-\frac {(e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{2 b d e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}-\frac {i e^{-2 i a d} (e x)^{1+m} \left (\frac {e^{2 i a d} (1+m-2 i b d n)}{n}-\frac {e^{4 i a d} (1+m+2 i b d n) \left (c x^n\right )^{2 i b d}}{n}\right )}{2 b^2 d^2 e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {i \left (1+2 m+m^2-2 b^2 d^2 n^2\right ) (e x)^{1+m} \, _2F_1\left (1,-\frac {i (1+m)}{2 b d n};1-\frac {i (1+m)}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b^2 d^2 e (1+m) n^2} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4593, 4591, 516, 608, 470, 371} \begin {gather*} \frac {i (e x)^{m+1} \left (-2 b^2 d^2 n^2+m^2+2 m+1\right ) \, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b^2 d^2 e (m+1) n^2}-\frac {i e^{-2 i a d} (e x)^{m+1} \left (\frac {e^{2 i a d} (-2 i b d n+m+1)}{n}-\frac {e^{4 i a d} (2 i b d n+m+1) \left (c x^n\right )^{2 i b d}}{n}\right )}{2 b^2 d^2 e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {(e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{2 b d e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}-\frac {(e x)^{m+1} (-b d n+i (m+1)) (2 i b d n+m+1)}{2 b^2 d^2 e (m+1) n^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^3,x]

[Out]

-1/2*((I*(1 + m) - b*d*n)*(1 + m + (2*I)*b*d*n)*(e*x)^(1 + m))/(b^2*d^2*e*(1 + m)*n^2) - ((e*x)^(1 + m)*(1 - E
^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^2)/(2*b*d*e*n*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^2) - ((I/2)*(e*x)^(1 +
m)*((E^((2*I)*a*d)*(1 + m - (2*I)*b*d*n))/n - (E^((4*I)*a*d)*(1 + m + (2*I)*b*d*n)*(c*x^n)^((2*I)*b*d))/n))/(
b^2*d^2*e*E^((2*I)*a*d)*n*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))) + (I*(1 + 2*m + m^2 - 2*b^2*d^2*n^2)*(e*x)^
(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^
((2*I)*b*d))])/(b^2*d^2*e*(1 + m)*n^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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 516

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d,
0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 608

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n},
x] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 4591

Int[((e_.)*(x_))^(m_.)*Tan[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(e*x)^m*((I - I*E^(2*I*a*d)*
x^(2*I*b*d))/(1 + E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]

Rule 4593

Int[((e_.)*(x_))^(m_.)*Tan[((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)*Tan[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 (e x)^m \tan ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \tan ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [A]
time = 18.01, size = 642, normalized size = 1.83 \begin {gather*} \frac {x (e x)^m \sec ^2\left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{2 b d n}-\frac {(1+m) x (e x)^m \sec \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \sec \left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \sin (b d n \log (x))}{2 b^2 d^2 n^2}-\frac {\left (-1-2 m-m^2+2 b^2 d^2 n^2\right ) x^{-m} (e x)^m \sec \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \left (\frac {x^{1+m} \sec \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))}{1+m}-\frac {i e^{-\frac {(1+2 m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \cos \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \left (-e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \, _2F_1\left (1,-\frac {i (1+m)}{2 b d n};1-\frac {i (1+m)}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+e^{\frac {a (1+2 m+2 i b d n)}{b n}+(1+m+2 i b d n) \log (x)+\frac {(1+2 m+2 i b d n) \left (-n \log (x)+\log \left (c x^n\right )\right )}{n}} (1+m) \, _2F_1\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n};-\frac {i (1+m+4 i b d n)}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-i e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{(1+m) (1+m+2 i b d n)}\right )}{2 b^2 d^2 n^2}-\frac {x (e x)^m \tan \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{1+m} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^3,x]

[Out]

(x*(e*x)^m*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]^2)/(2*b*d*n) - ((1 + m)*x*(e*x)^m*Sec[d*(a
+ b*(-(n*Log[x]) + Log[c*x^n]))]*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/
(2*b^2*d^2*n^2) - ((-1 - 2*m - m^2 + 2*b^2*d^2*n^2)*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 +
m)*Sec[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*(-(E^(
(a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hyperge
ometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + E^(
(a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log
[c*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b
*d*n))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log
[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Tan[d*(a + b*Log[c*x^n])]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]
) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(2*b^2*d^2*n^2*x^m) - (x*(e*x)^m*Tan[d*(a + b*(-(n*L
og[x]) + Log[c*x^n]))])/(1 + m)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*tan(d*(a+b*ln(c*x^n)))^3,x)

[Out]

int((e*x)^m*tan(d*(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((e*x)^m*tan(d*(a+b*log(c*x^n)))^3,x, algorithm="maxima")

[Out]

(4*(b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*x*cos(2*b*d*log(x^n) + 2*a*d)^2*e^(m*log(x) + m) + 4*
(b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*x*e^(m*log(x) + m)*sin(2*b*d*log(x^n) + 2*a*d)^2 + (2*b*
d*n*cos(2*b*d*log(c))*e^m - (m*sin(2*b*d*log(c)) + sin(2*b*d*log(c)))*e^m)*x*x^m*cos(2*b*d*log(x^n) + 2*a*d) -
(2*b*d*n*e^m*sin(2*b*d*log(c)) + (m*cos(2*b*d*log(c)) + cos(2*b*d*log(c)))*e^m)*x*x^m*sin(2*b*d*log(x^n) + 2*
a*d) + ((2*(b*d*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*n*e^m - ((cos(2
*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(2*b*d*log(c)))*m + cos(2*b*d*log(c))*sin(4*b*d*log(c))
- cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*x*x^m*cos(2*b*d*log(x^n) + 2*a*d) + (2*(b*d*cos(2*b*d*log(c))*sin(
4*b*d*log(c)) - b*d*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*n*e^m + ((cos(4*b*d*log(c))*cos(2*b*d*log(c)) + sin(4
*b*d*log(c))*sin(2*b*d*log(c)))*m + cos(4*b*d*log(c))*cos(2*b*d*log(c)) + sin(4*b*d*log(c))*sin(2*b*d*log(c)))
*e^m)*x*x^m*sin(2*b*d*log(x^n) + 2*a*d) - (m*sin(4*b*d*log(c)) + sin(4*b*d*log(c)))*x*e^(m*log(x) + m))*cos(4*
b*d*log(x^n) + 4*a*d) - (2*b^6*d^6*n^6*e^m - (b^4*d^4*m^2 + 2*b^4*d^4*m + b^4*d^4)*n^4*e^m + (2*(b^6*d^6*cos(4
*b*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*n^6*e^m - (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c
))^2 + (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*
d^4*sin(4*b*d*log(c))^2)*m)*n^4*e^m)*cos(4*b*d*log(x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d
^6*sin(2*b*d*log(c))^2)*n^6*e^m - (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2 + (b^4*d^4*cos(2*
b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^
2)*m)*n^4*e^m)*cos(2*b*d*log(x^n) + 2*a*d)^2 + (2*(b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*
n^6*e^m - (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2 + (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*
sin(4*b*d*log(c))^2)*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*m)*n^4*e^m)*sin(4*b*d
*log(x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*n^6*e^m - (b^4*d^4*cos
(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2 + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*m
^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*m)*n^4*e^m)*sin(2*b*d*log(x^n) + 2*a*d)^2 +
2*(2*b^6*d^6*n^6*cos(4*b*d*log(c))*e^m - (b^4*d^4*m^2*cos(4*b*d*log(c)) + 2*b^4*d^4*m*cos(4*b*d*log(c)) + b^4
*d^4*cos(4*b*d*log(c)))*n^4*e^m + 2*(2*(b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c)
)*sin(2*b*d*log(c)))*n^6*e^m - (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*
b*d*log(c)) + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*m^2
+ 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*m)*n^4*e^m)*co
s(2*b*d*log(x^n) + 2*a*d) + 2*(2*(b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(
2*b*d*log(c)))*n^6*e^m - (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*lo
g(c)) + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*m^2 + 2*(b
^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*m)*n^4*e^m)*sin(2*b*
d*log(x^n) + 2*a*d))*cos(4*b*d*log(x^n) + 4*a*d) + 4*(2*b^6*d^6*n^6*cos(2*b*d*log(c))*e^m - (b^4*d^4*m^2*cos(2
*b*d*log(c)) + 2*b^4*d^4*m*cos(2*b*d*log(c)) + b^4*d^4*cos(2*b*d*log(c)))*n^4*e^m)*cos(2*b*d*log(x^n) + 2*a*d)
- 2*(2*b^6*d^6*n^6*e^m*sin(4*b*d*log(c)) - (b^4*d^4*m^2*sin(4*b*d*log(c)) + 2*b^4*d^4*m*sin(4*b*d*log(c)) + b
^4*d^4*sin(4*b*d*log(c)))*n^4*e^m + 2*(2*(b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(
c))*sin(2*b*d*log(c)))*n^6*e^m - (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(
2*b*d*log(c)) + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*m^
2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*m)*n^4*e^m)*
cos(2*b*d*log(x^n) + 2*a*d) - 2*(2*(b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*si
n(2*b*d*log(c)))*n^6*e^m - (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*
log(c)) + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*m^2 + 2*
(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*m)*n^4*e^m)*sin(2*
b*d*log(x^n) + 2*a*d))*sin(4*b*d*log(x^n) + 4*a*d) - 4*(2*b^6*d^6*n^6*e^m*sin(2*b*d*log(c)) - (b^4*d^4*m^2*sin
(2*b*d*log(c)) + 2*b^4*d^4*m*sin(2*b*d*log(c)) + b^4*d^4*sin(2*b*d*log(c)))*n^4*e^m)*sin(2*b*d*log(x^n) + 2*a*
d))*integrate((x^m*cos(2*b*d*log(x^n) + 2*a*d)*sin(2*b*d*log(c)) + x^m*cos(2*b*d*log(c))*sin(2*b*d*log(x^n) +
2*a*d))/(2*b^4*d^4*n^4*cos(2*b*d*log(c))*cos(2*...

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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((e*x)^m*tan(d*(a+b*log(c*x^n)))^3,x, algorithm="fricas")

[Out]

integral((x*e)^m*tan(b*d*log(c*x^n) + a*d)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{m} \tan ^{3}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*tan(d*(a+b*ln(c*x**n)))**3,x)

[Out]

Integral((e*x)**m*tan(a*d + b*d*log(c*x**n))**3, 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((e*x)^m*tan(d*(a+b*log(c*x^n)))^3,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^3\,{\left (e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*(a + b*log(c*x^n)))^3*(e*x)^m,x)

[Out]

int(tan(d*(a + b*log(c*x^n)))^3*(e*x)^m, x)

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