3.414 \(\int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=60 \[ \frac{a \tan ^2(c+d x)}{2 d}+\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x \]

[Out]

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

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Rubi [A]  time = 0.0581332, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac{a \tan ^2(c+d x)}{2 d}+\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x + (a*Log[Cos[c + d*x]])/d - (b*Tan[c + d*x])/d + (a*Tan[c + d*x]^2)/(2*d) + (b*Tan[c + d*x]^3)/(3*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 ^3(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{b \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac{a \tan ^2(c+d x)}{2 d}+\frac{b \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-a-b \tan (c+d x)) \, dx\\ &=b x-\frac{b \tan (c+d x)}{d}+\frac{a \tan ^2(c+d x)}{2 d}+\frac{b \tan ^3(c+d x)}{3 d}-a \int \tan (c+d x) \, dx\\ &=b x+\frac{a \log (\cos (c+d x))}{d}-\frac{b \tan (c+d x)}{d}+\frac{a \tan ^2(c+d x)}{2 d}+\frac{b \tan ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.127224, size = 67, normalized size = 1.12 \[ \frac{a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d}+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan ^{-1}(\tan (c+d x))}{d}-\frac{b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 71, normalized size = 1.2 \begin{align*}{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{b\tan \left ( dx+c \right ) }{d}}-{\frac{a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+{\frac{b\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)^3*(a+b*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.62725, size = 80, normalized size = 1.33 \begin{align*} \frac{2 \, b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )^{2} + 6 \,{\left (d x + c\right )} b - 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58741, size = 151, normalized size = 2.52 \begin{align*} \frac{2 \, b \tan \left (d x + c\right )^{3} + 6 \, b d x + 3 \, a \tan \left (d x + c\right )^{2} + 3 \, a \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.372844, size = 70, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \tan ^{2}{\left (c + d x \right )}}{2 d} + b x + \frac{b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{b \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \tan ^{3}{\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)**3*(a+b*tan(d*x+c)),x)

[Out]

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

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Giac [B]  time = 2.26677, size = 695, normalized size = 11.58 \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)^3*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*b*d*x*tan(d*x)^3*tan(c)^3 + 3*a*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 - 18*b*d*x*tan(d*x)^2*tan(c)^2 + 3*a
*tan(d*x)^3*tan(c)^3 - 9*a*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 + 6*b*tan(d*x)^3*tan(c)^2 + 6*b*tan(d*x)^2*tan(c
)^3 + 18*b*d*x*tan(d*x)*tan(c) + 3*a*tan(d*x)^3*tan(c) - 3*a*tan(d*x)^2*tan(c)^2 + 3*a*tan(d*x)*tan(c)^3 - 2*b
*tan(d*x)^3 + 9*a*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)*tan(c) - 18*b*tan(d*x)^2*tan(c) - 18*b*tan(d*x)*tan(c)^2 - 2*b*tan(c
)^3 - 6*b*d*x - 3*a*tan(d*x)^2 + 3*a*tan(d*x)*tan(c) - 3*a*tan(c)^2 - 3*a*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)) + 6*b*tan(d*x) + 6*b*
tan(c) - 3*a)/(d*tan(d*x)^3*tan(c)^3 - 3*d*tan(d*x)^2*tan(c)^2 + 3*d*tan(d*x)*tan(c) - d)