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

Optimal. Leaf size=77 \[ \frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+a x+\frac{b \tan ^4(c+d x)}{4 d}-\frac{b \tan ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d} \]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d - (a*Tan[c + d*x])/d - (b*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^3)/(3*d) + (b*
Tan[c + d*x]^4)/(4*d)

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Rubi [A]  time = 0.0785955, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+a x+\frac{b \tan ^4(c+d x)}{4 d}-\frac{b \tan ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4*(a + b*Tan[c + d*x]),x]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d - (a*Tan[c + d*x])/d - (b*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^3)/(3*d) + (b*
Tan[c + d*x]^4)/(4*d)

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3525

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.193612, size = 79, normalized size = 1.03 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^{-1}(\tan (c+d x))}{d}-\frac{a \tan (c+d x)}{d}-\frac{b \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x]),x]

[Out]

(a*ArcTan[Tan[c + d*x]])/d - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d) - (b*(4*Log[Cos[c + d*x]] + 2*Tan[c
+ d*x]^2 - Tan[c + d*x]^4))/(4*d)

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Maple [A]  time = 0.003, size = 85, normalized size = 1.1 \begin{align*}{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\tan \left ( dx+c \right ) }{d}}+{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+{\frac{a\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*tan(d*x+c)^4/d+1/3*a*tan(d*x+c)^3/d-1/2*b*tan(d*x+c)^2/d-a*tan(d*x+c)/d+1/2/d*b*ln(1+tan(d*x+c)^2)+1/d*a
*arctan(tan(d*x+c))

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Maxima [A]  time = 1.69881, size = 95, normalized size = 1.23 \begin{align*} \frac{3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} - 6 \, b \tan \left (d x + c\right )^{2} + 12 \,{\left (d x + c\right )} a + 6 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*tan(d*x + c)^4 + 4*a*tan(d*x + c)^3 - 6*b*tan(d*x + c)^2 + 12*(d*x + c)*a + 6*b*log(tan(d*x + c)^2 +
1) - 12*a*tan(d*x + c))/d

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Fricas [A]  time = 1.6946, size = 184, normalized size = 2.39 \begin{align*} \frac{3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 12 \, a d x - 6 \, b \tan \left (d x + c\right )^{2} - 6 \, b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*tan(d*x + c)^4 + 4*a*tan(d*x + c)^3 + 12*a*d*x - 6*b*tan(d*x + c)^2 - 6*b*log(1/(tan(d*x + c)^2 + 1)
) - 12*a*tan(d*x + c))/d

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Sympy [A]  time = 0.540554, size = 83, normalized size = 1.08 \begin{align*} \begin{cases} a x + \frac{a \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{a \tan{\left (c + d x \right )}}{d} + \frac{b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \tan ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x + a*tan(c + d*x)**3/(3*d) - a*tan(c + d*x)/d + b*log(tan(c + d*x)**2 + 1)/(2*d) + b*tan(c + d*x
)**4/(4*d) - b*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))*tan(c)**4, True))

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Giac [B]  time = 3.4699, size = 967, normalized size = 12.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*a*d*x*tan(d*x)^4*tan(c)^4 - 6*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan
(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 - 48*a*d*x*tan(d*x)^3*tan(c)^3 - 9
*b*tan(d*x)^4*tan(c)^4 + 24*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan
(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 + 12*a*tan(d*x)^4*tan(c)^3 + 12*a*tan(d*x)^3*
tan(c)^4 + 72*a*d*x*tan(d*x)^2*tan(c)^2 - 6*b*tan(d*x)^4*tan(c)^2 + 24*b*tan(d*x)^3*tan(c)^3 - 6*b*tan(d*x)^2*
tan(c)^4 - 4*a*tan(d*x)^4*tan(c) - 36*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(
d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 - 48*a*tan(d*x)^3*tan(c)^2 - 48*a*t
an(d*x)^2*tan(c)^3 - 4*a*tan(d*x)*tan(c)^4 + 3*b*tan(d*x)^4 - 48*a*d*x*tan(d*x)*tan(c) + 24*b*tan(d*x)^3*tan(c
) - 12*b*tan(d*x)^2*tan(c)^2 + 24*b*tan(d*x)*tan(c)^3 + 3*b*tan(c)^4 + 4*a*tan(d*x)^3 + 24*b*log(4*(tan(c)^2 +
1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*ta
n(d*x)*tan(c) + 48*a*tan(d*x)^2*tan(c) + 48*a*tan(d*x)*tan(c)^2 + 4*a*tan(c)^3 + 12*a*d*x - 6*b*tan(d*x)^2 + 2
4*b*tan(d*x)*tan(c) - 6*b*tan(c)^2 - 6*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan
(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 12*a*tan(d*x) - 12*a*tan(c) - 9*b)/(d*tan(d*x)^4*tan
(c)^4 - 4*d*tan(d*x)^3*tan(c)^3 + 6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)