3.1.19 \(\int e^{c (a+b x)} \tan ^3(d+e x) \, dx\) [19]
Optimal. Leaf size=194 \[ \frac {i e^{c (a+b x)}}{b c}-\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c} \]
[Out]
I*exp(c*(b*x+a))/b/c-6*I*exp(c*(b*x+a))*hypergeom([1, -1/2*I*b*c/e],[1-1/2*I*b*c/e],-exp(2*I*(e*x+d)))/b/c+12*
I*exp(c*(b*x+a))*hypergeom([2, -1/2*I*b*c/e],[1-1/2*I*b*c/e],-exp(2*I*(e*x+d)))/b/c-8*I*exp(c*(b*x+a))*hyperge
om([3, -1/2*I*b*c/e],[1-1/2*I*b*c/e],-exp(2*I*(e*x+d)))/b/c
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Rubi [A]
time = 0.15, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4527, 2225,
2283} \begin {gather*} -\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {i e^{c (a+b x)}}{b c} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[E^(c*(a + b*x))*Tan[d + e*x]^3,x]
[Out]
(I*E^(c*(a + b*x)))/(b*c) - ((6*I)*E^(c*(a + b*x))*Hypergeometric2F1[1, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e, -
E^((2*I)*(d + e*x))])/(b*c) + ((12*I)*E^(c*(a + b*x))*Hypergeometric2F1[2, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e
, -E^((2*I)*(d + e*x))])/(b*c) - ((8*I)*E^(c*(a + b*x))*Hypergeometric2F1[3, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)
/e, -E^((2*I)*(d + e*x))])/(b*c)
Rule 2225
Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]
Rule 2283
Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])
Rule 4527
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Dist[I^n, Int[ExpandIntegran
d[F^(c*(a + b*x))*((1 - E^(2*I*(d + e*x)))^n/(1 + E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e
}, x] && IntegerQ[n]
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tan ^3(d+e x) \, dx &=-\left (i \int \left (-e^{c (a+b x)}+\frac {8 e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^3}-\frac {12 e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^2}+\frac {6 e^{c (a+b x)}}{1+e^{2 i (d+e x)}}\right ) \, dx\right )\\ &=i \int e^{c (a+b x)} \, dx-6 i \int \frac {e^{c (a+b x)}}{1+e^{2 i (d+e x)}} \, dx-8 i \int \frac {e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^3} \, dx+12 i \int \frac {e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^2} \, dx\\ &=\frac {i e^{c (a+b x)}}{b c}-\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}\\ \end {align*}
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Mathematica [A]
time = 2.17, size = 212, normalized size = 1.09 \begin {gather*} \frac {1}{2} e^{c (a+b x)} \left (\frac {2 \left (b^2 c^2-2 e^2\right ) e^{2 i d} \left (b c e^{2 i e x} \, _2F_1\left (1,1-\frac {i b c}{2 e};2-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )-(b c+2 i e) \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )\right )}{b c (i b c-2 e) e^2 \left (1+e^{2 i d}\right )}+\frac {\sec ^2(d+e x)}{e}-\frac {b c \sec (d) \sec (d+e x) \sin (e x)}{e^2}-\frac {2 \tan (d)}{b c}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[E^(c*(a + b*x))*Tan[d + e*x]^3,x]
[Out]
(E^(c*(a + b*x))*((2*(b^2*c^2 - 2*e^2)*E^((2*I)*d)*(b*c*E^((2*I)*e*x)*Hypergeometric2F1[1, 1 - ((I/2)*b*c)/e,
2 - ((I/2)*b*c)/e, -E^((2*I)*(d + e*x))] - (b*c + (2*I)*e)*Hypergeometric2F1[1, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b
*c)/e, -E^((2*I)*(d + e*x))]))/(b*c*(I*b*c - 2*e)*e^2*(1 + E^((2*I)*d))) + Sec[d + e*x]^2/e - (b*c*Sec[d]*Sec[
d + e*x]*Sin[e*x])/e^2 - (2*Tan[d])/(b*c)))/2
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{c \left (b x +a \right )} \left (\tan ^{3}\left (e x +d \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(exp(c*(b*x+a))*tan(e*x+d)^3,x)
[Out]
int(exp(c*(b*x+a))*tan(e*x+d)^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(exp(c*(b*x+a))*tan(e*x+d)^3,x, algorithm="maxima")
[Out]
2*(18*(b^4*c^4*e + 52*b^2*c^2*e^3 + 576*e^5)*cos(4*x*e + 4*d)^2*e^(b*c*x + a*c) - 54*(b^4*c^4*e + 28*b^2*c^2*e
^3 - 288*e^5)*cos(2*x*e + 2*d)^2*e^(b*c*x + a*c) + 18*(b^4*c^4*e + 52*b^2*c^2*e^3 + 576*e^5)*e^(b*c*x + a*c)*s
in(4*x*e + 4*d)^2 - 54*(b^4*c^4*e + 28*b^2*c^2*e^3 - 288*e^5)*e^(b*c*x + a*c)*sin(2*x*e + 2*d)^2 + 18*(3*b^4*c
^4*e - 212*b^2*c^2*e^3 + 640*e^5)*cos(2*x*e + 2*d)*e^(b*c*x + a*c) - 3*(b^5*c^5 - 268*b^3*c^3*e^2 + 1216*b*c*e
^4)*e^(b*c*x + a*c)*sin(2*x*e + 2*d) + 3*(2*(b^4*c^4*e + 52*b^2*c^2*e^3 + 576*e^5)*cos(4*x*e + 4*d)*e^(b*c*x +
a*c) - 6*(b^4*c^4*e + 28*b^2*c^2*e^3 - 288*e^5)*cos(2*x*e + 2*d)*e^(b*c*x + a*c) + (b^5*c^5 + 52*b^3*c^3*e^2
+ 576*b*c*e^4)*e^(b*c*x + a*c)*sin(4*x*e + 4*d) + 36*(b^3*c^3*e^2 + 36*b*c*e^4)*e^(b*c*x + a*c)*sin(2*x*e + 2*
d) + 8*(b^4*c^4*e - 46*b^2*c^2*e^3 + 88*e^5)*e^(b*c*x + a*c))*cos(6*x*e + 6*d) - 3*(12*(b^4*c^4*e + 16*b^2*c^2
*e^3 - 720*e^5)*cos(2*x*e + 2*d)*e^(b*c*x + a*c) + 3*(b^5*c^5 + 16*b^3*c^3*e^2 - 720*b*c*e^4)*e^(b*c*x + a*c)*
sin(2*x*e + 2*d) - 2*(13*b^4*c^4*e - 500*b^2*c^2*e^3 + 1632*e^5)*e^(b*c*x + a*c))*cos(4*x*e + 4*d) + 24*(b^4*c
^4*e - 46*b^2*c^2*e^3 + 88*e^5)*e^(b*c*x + a*c) - 24*((b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*
c^2*e^7 - 4608*e^9)*cos(6*x*e + 6*d)^2*e^(a*c) + 9*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2
*e^7 - 4608*e^9)*cos(4*x*e + 4*d)^2*e^(a*c) + 9*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^
7 - 4608*e^9)*cos(2*x*e + 2*d)^2*e^(a*c) + (b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4
608*e^9)*e^(a*c)*sin(6*x*e + 6*d)^2 + 9*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608
*e^9)*e^(a*c)*sin(4*x*e + 4*d)^2 + 18*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e
^9)*e^(a*c)*sin(4*x*e + 4*d)*sin(2*x*e + 2*d) + 9*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*
e^7 - 4608*e^9)*e^(a*c)*sin(2*x*e + 2*d)^2 + 6*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7
- 4608*e^9)*cos(2*x*e + 2*d)*e^(a*c) + 2*(3*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 -
4608*e^9)*cos(4*x*e + 4*d)*e^(a*c) + 3*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608
*e^9)*cos(2*x*e + 2*d)*e^(a*c) + (b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*e
^(a*c))*cos(6*x*e + 6*d) + 6*(3*(b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*co
s(2*x*e + 2*d)*e^(a*c) + (b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*e^(a*c))*
cos(4*x*e + 4*d) + (b^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*e^(a*c) + 6*((b
^8*c^8*e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*e^(a*c)*sin(4*x*e + 4*d) + (b^8*c^8*
e + 54*b^6*c^6*e^3 + 672*b^4*c^4*e^5 + 736*b^2*c^2*e^7 - 4608*e^9)*e^(a*c)*sin(2*x*e + 2*d))*sin(6*x*e + 6*d))
*integrate(((b^3*c^3 - 44*b*c*e^2)*cos(8*x*e + 8*d)*e^(b*c*x) + 4*(b^3*c^3 - 44*b*c*e^2)*cos(6*x*e + 6*d)*e^(b
*c*x) + 6*(b^3*c^3 - 44*b*c*e^2)*cos(4*x*e + 4*d)*e^(b*c*x) + 4*(b^3*c^3 - 44*b*c*e^2)*cos(2*x*e + 2*d)*e^(b*c
*x) + 12*(b^2*c^2*e - 4*e^3)*e^(b*c*x)*sin(8*x*e + 8*d) + 48*(b^2*c^2*e - 4*e^3)*e^(b*c*x)*sin(6*x*e + 6*d) +
72*(b^2*c^2*e - 4*e^3)*e^(b*c*x)*sin(4*x*e + 4*d) + 48*(b^2*c^2*e - 4*e^3)*e^(b*c*x)*sin(2*x*e + 2*d) + (b^3*c
^3 - 44*b*c*e^2)*e^(b*c*x))/(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + (b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*
c^2*e^4 + 2304*e^6)*cos(8*x*e + 8*d)^2 + 16*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(6*x*e
+ 6*d)^2 + 36*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(4*x*e + 4*d)^2 + 16*(b^6*c^6 + 56*b^
4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(2*x*e + 2*d)^2 + (b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 230
4*e^6)*sin(8*x*e + 8*d)^2 + 16*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*sin(6*x*e + 6*d)^2 + 36
*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*sin(4*x*e + 4*d)^2 + 48*(b^6*c^6 + 56*b^4*c^4*e^2 + 7
84*b^2*c^2*e^4 + 2304*e^6)*sin(4*x*e + 4*d)*sin(2*x*e + 2*d) + 16*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4
+ 2304*e^6)*sin(2*x*e + 2*d)^2 + 2*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 4*(b^6*c^6 + 56*b^4*c^4*e^2 +
784*b^2*c^2*e^4 + 2304*e^6)*cos(6*x*e + 6*d) + 6*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(
4*x*e + 4*d) + 4*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(2*x*e + 2*d) + 2304*e^6)*cos(8*x*
e + 8*d) + 8*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 6*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 230
4*e^6)*cos(4*x*e + 4*d) + 4*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*cos(2*x*e + 2*d) + 2304*e^
6)*cos(6*x*e + 6*d) + 12*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 4*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c
^2*e^4 + 2304*e^6)*cos(2*x*e + 2*d) + 2304*e^6)*cos(4*x*e + 4*d) + 8*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e
^4 + 2304*e^6)*cos(2*x*e + 2*d) + 4*(2*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e^4 + 2304*e^6)*sin(6*x*e + 6*d
) + 3*(b^6*c^6 + 56*b^4*c^4*e^2 + 784*b^2*c^2*e...
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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(exp(c*(b*x+a))*tan(e*x+d)^3,x, algorithm="fricas")
[Out]
integral(e^(b*c*x + a*c)*tan(x*e + d)^3, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a c} \int e^{b c x} \tan ^{3}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(exp(c*(b*x+a))*tan(e*x+d)**3,x)
[Out]
exp(a*c)*Integral(exp(b*c*x)*tan(d + e*x)**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(exp(c*(b*x+a))*tan(e*x+d)^3,x, algorithm="giac")
[Out]
integrate(e^((b*x + a)*c)*tan(e*x + d)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\mathrm {tan}\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(exp(c*(a + b*x))*tan(d + e*x)^3,x)
[Out]
int(exp(c*(a + b*x))*tan(d + e*x)^3, x)
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