3.298 \(\int \frac{\tan ^3(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{B \tan ^2(c+d x)}{2 d}+\frac{B \log (\cos (c+d x))}{d} \]

[Out]

(B*Log[Cos[c + d*x]])/d + (B*Tan[c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.0175713, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {21, 3473, 3475} \[ \frac{B \tan ^2(c+d x)}{2 d}+\frac{B \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*Log[Cos[c + d*x]])/d + (B*Tan[c + d*x]^2)/(2*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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]

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

Mathematica [A]  time = 0.028094, size = 26, normalized size = 0.9 \[ \frac{B \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)

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Maple [A]  time = 0.022, size = 33, normalized size = 1.1 \begin{align*}{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

1/2*B*tan(d*x+c)^2/d-1/2/d*B*ln(1+tan(d*x+c)^2)

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Maxima [A]  time = 1.48099, size = 41, normalized size = 1.41 \begin{align*} \frac{B \tan \left (d x + c\right )^{2} - B \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*tan(d*x + c)^2 - B*log(tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.71534, size = 78, normalized size = 2.69 \begin{align*} \frac{B \tan \left (d x + c\right )^{2} + B \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*tan(d*x + c)^2 + B*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 14.0065, size = 53, normalized size = 1.83 \begin{align*} \begin{cases} - \frac{B \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \tan{\left (c \right )}\right ) \tan ^{3}{\left (c \right )}}{a + b \tan{\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+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

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

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Giac [B]  time = 1.83875, size = 252, normalized size = 8.69 \begin{align*} -\frac{B \log \left ({\left | -\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 2 \right |}\right ) - B \log \left ({\left | -\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} - \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 2 \right |}\right ) + \frac{B{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 6 \, B}{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 2}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*log(abs(-(cos(d*x + c) + 1)/(cos(d*x + c) - 1) - (cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2)) - B*log(a
bs(-(cos(d*x + c) + 1)/(cos(d*x + c) - 1) - (cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2)) + (B*((cos(d*x + c) +
1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 6*B)/((cos(d*x + c) + 1)/(cos(d*x + c) - 1) +
(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2))/d