3.425 \(\int \tan ^3(c+d x) (a+b \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=98 \[ \frac{\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 d}+\frac{\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}-\frac{2 a b \tan (c+d x)}{d}+2 a b x+\frac{b^2 \tan ^4(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.127518, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3543, 3528, 3525, 3475} \[ \frac{\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 d}+\frac{\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}-\frac{2 a b \tan (c+d x)}{d}+2 a b x+\frac{b^2 \tan ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]

[Out]

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

Rule 3543

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

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

Mathematica [A]  time = 0.432209, size = 113, normalized size = 1.15 \[ \frac{a^2 \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan ^{-1}(\tan (c+d x))}{d}-\frac{2 a b \tan (c+d x)}{d}-\frac{b^2 \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]

[Out]

(2*a*b*ArcTan[Tan[c + d*x]])/d - (2*a*b*Tan[c + d*x])/d + (2*a*b*Tan[c + d*x]^3)/(3*d) + (a^2*(2*Log[Cos[c + d
*x]] + Tan[c + d*x]^2))/(2*d) - (b^2*(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 = 130, normalized size = 1.3 \begin{align*}{\frac{{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{2\,ab \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{2}}{2\,d}}-2\,{\frac{ab\tan \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{2}}{2\,d}}+2\,{\frac{ab\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))^2,x)

[Out]

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

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Maxima [A]  time = 1.53824, size = 123, normalized size = 1.26 \begin{align*} \frac{3 \, b^{2} \tan \left (d x + c\right )^{4} + 8 \, a b \tan \left (d x + c\right )^{3} + 24 \,{\left (d x + c\right )} a b - 24 \, a b \tan \left (d x + c\right ) + 6 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \,{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{12 \, 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))^2,x, algorithm="maxima")

[Out]

1/12*(3*b^2*tan(d*x + c)^4 + 8*a*b*tan(d*x + c)^3 + 24*(d*x + c)*a*b - 24*a*b*tan(d*x + c) + 6*(a^2 - b^2)*tan
(d*x + c)^2 - 6*(a^2 - b^2)*log(tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.79812, size = 221, normalized size = 2.26 \begin{align*} \frac{3 \, b^{2} \tan \left (d x + c\right )^{4} + 8 \, a b \tan \left (d x + c\right )^{3} + 24 \, a b d x - 24 \, a b \tan \left (d x + c\right ) + 6 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + 6 \,{\left (a^{2} - b^{2}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{12 \, 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))^2,x, algorithm="fricas")

[Out]

1/12*(3*b^2*tan(d*x + c)^4 + 8*a*b*tan(d*x + c)^3 + 24*a*b*d*x - 24*a*b*tan(d*x + c) + 6*(a^2 - b^2)*tan(d*x +
c)^2 + 6*(a^2 - b^2)*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 0.631458, size = 134, normalized size = 1.37 \begin{align*} \begin{cases} - \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + 2 a b x + \frac{2 a b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{2 a b \tan{\left (c + d x \right )}}{d} + \frac{b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{2} \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))**2,x)

[Out]

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

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Giac [B]  time = 3.8884, size = 1704, normalized size = 17.39 \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))^2,x, algorithm="giac")

[Out]

1/12*(24*a*b*d*x*tan(d*x)^4*tan(c)^4 + 6*a^2*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 - 6*b^2*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 - 96*a*b*d*x*tan(d*x)^3*tan(c)^3 + 6*a^2*tan(d*x)^4*tan(c)^4 - 9*b^2*tan(d*x)^4*tan(c)^4 - 24*a^2
*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 + 24*b^2*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 + 24*a*b*tan(d*x)^4*tan(c)^3 +
24*a*b*tan(d*x)^3*tan(c)^4 + 144*a*b*d*x*tan(d*x)^2*tan(c)^2 + 6*a^2*tan(d*x)^4*tan(c)^2 - 6*b^2*tan(d*x)^4*t
an(c)^2 - 12*a^2*tan(d*x)^3*tan(c)^3 + 24*b^2*tan(d*x)^3*tan(c)^3 + 6*a^2*tan(d*x)^2*tan(c)^4 - 6*b^2*tan(d*x)
^2*tan(c)^4 - 8*a*b*tan(d*x)^4*tan(c) + 36*a^2*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 - 36*b^2*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 - 96*a*b*tan(d*x)^3*tan(c)^2 - 96*a*b*tan(d*x)^2*tan(c)^3 - 8*a*b*tan(d*x)*tan(c)^4 + 3*b^2*ta
n(d*x)^4 - 96*a*b*d*x*tan(d*x)*tan(c) - 12*a^2*tan(d*x)^3*tan(c) + 24*b^2*tan(d*x)^3*tan(c) + 12*a^2*tan(d*x)^
2*tan(c)^2 - 12*b^2*tan(d*x)^2*tan(c)^2 - 12*a^2*tan(d*x)*tan(c)^3 + 24*b^2*tan(d*x)*tan(c)^3 + 3*b^2*tan(c)^4
+ 8*a*b*tan(d*x)^3 - 24*a^2*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) + 24*b^2*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) + 96*a*b*t
an(d*x)^2*tan(c) + 96*a*b*tan(d*x)*tan(c)^2 + 8*a*b*tan(c)^3 + 24*a*b*d*x + 6*a^2*tan(d*x)^2 - 6*b^2*tan(d*x)^
2 - 12*a^2*tan(d*x)*tan(c) + 24*b^2*tan(d*x)*tan(c) + 6*a^2*tan(c)^2 - 6*b^2*tan(c)^2 + 6*a^2*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^2*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)) - 24*a*b*tan(d*x) - 24*a*b*tan(c) + 6*a^2 - 9*b^2)/(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)