3.384 \(\int (a+b \tan ^4(c+d x)) \, dx\)

Optimal. Leaf size=35 \[ a x+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]

[Out]

a*x+b*x-b*tan(d*x+c)/d+1/3*b*tan(d*x+c)^3/d

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Rubi [A]  time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3473, 8} \[ a x+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Tan[c + d*x]^4,x]

[Out]

a*x + b*x - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \left (a+b \tan ^4(c+d x)\right ) \, dx &=a x+b \int \tan ^4(c+d x) \, dx\\ &=a x+\frac {b \tan ^3(c+d x)}{3 d}-b \int \tan ^2(c+d x) \, dx\\ &=a x-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d}+b \int 1 \, dx\\ &=a x+b x-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 1.26 \[ a x+\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Tan[c + d*x]^4,x]

[Out]

a*x + (b*ArcTan[Tan[c + d*x]])/d - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)

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fricas [A]  time = 0.64, size = 32, normalized size = 0.91 \[ \frac {b \tan \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} d x - 3 \, b \tan \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+tan(d*x+c)^4*b,x, algorithm="fricas")

[Out]

1/3*(b*tan(d*x + c)^3 + 3*(a + b)*d*x - 3*b*tan(d*x + c))/d

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+tan(d*x+c)^4*b,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)b*(12*d*x*tan(c)^3
*tan(d*x)^3-36*d*x*tan(c)^2*tan(d*x)^2+36*d*x*tan(c)*tan(d*x)-12*d*x-3*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*ta
n(d*x)^2-2*tan(c)-2*tan(d*x))*tan(c)^3*tan(d*x)^3+9*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2
*tan(d*x))*tan(c)^2*tan(d*x)^2-9*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))*tan(c)*t
an(d*x)+3*pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))-3*pi*tan(c)^3*tan(d*x)^3+9*pi*t
an(c)^2*tan(d*x)^2-9*pi*tan(c)*tan(d*x)+3*pi+6*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)^3*tan(d*x)^3
-18*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)^2*tan(d*x)^2+18*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*
x)))*tan(c)*tan(d*x)-6*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))+6*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1
))*tan(c)^3*tan(d*x)^3-18*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)^2*tan(d*x)^2+18*atan((tan(c)+tan(
d*x))/(tan(c)*tan(d*x)-1))*tan(c)*tan(d*x)-6*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))+12*tan(c)^3*tan(d*x)^
2-4*tan(c)^3+12*tan(c)^2*tan(d*x)^3-36*tan(c)^2*tan(d*x)-36*tan(c)*tan(d*x)^2+12*tan(c)-4*tan(d*x)^3+12*tan(d*
x))/(12*d*tan(c)^3*tan(d*x)^3-36*d*tan(c)^2*tan(d*x)^2+36*d*tan(c)*tan(d*x)-12*d)+a*x

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maple [A]  time = 0.02, size = 43, normalized size = 1.23 \[ a x +\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b \tan \left (d x +c \right )}{d}+\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*tan(d*x+c)^4,x)

[Out]

a*x+1/3*b*tan(d*x+c)^3/d-b*tan(d*x+c)/d+1/d*b*arctan(tan(d*x+c))

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maxima [A]  time = 0.46, size = 34, normalized size = 0.97 \[ a x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} b}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+tan(d*x+c)^4*b,x, algorithm="maxima")

[Out]

a*x + 1/3*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*b/d

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mupad [B]  time = 11.59, size = 31, normalized size = 0.89 \[ \frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-b\,\mathrm {tan}\left (c+d\,x\right )+d\,x\,\left (a+b\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*tan(c + d*x)^4,x)

[Out]

((b*tan(c + d*x)^3)/3 - b*tan(c + d*x) + d*x*(a + b))/d

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sympy [A]  time = 0.22, size = 32, normalized size = 0.91 \[ a x + b \left (\begin {cases} x + \frac {\tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{4}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+tan(d*x+c)**4*b,x)

[Out]

a*x + b*Piecewise((x + tan(c + d*x)**3/(3*d) - tan(c + d*x)/d, Ne(d, 0)), (x*tan(c)**4, True))

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