3.52 \(\int (b \tan ^p(c+d x))^{\frac{1}{p}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*(b*Tan[c + d*x]^p)^p^(-1))/d)

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Rubi [A]  time = 0.0191736, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3659, 3475} \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x]^p)^p^(-1),x]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*(b*Tan[c + d*x]^p)^p^(-1))/d)

Rule 3659

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x
])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

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 \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}} \, dx &=\left (\cot (c+d x) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d}\\ \end{align*}

Mathematica [A]  time = 0.0220582, size = 32, normalized size = 1. \[ -\frac{\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac{1}{p}}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x]^p)^p^(-1),x]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*(b*Tan[c + d*x]^p)^p^(-1))/d)

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Maple [C]  time = 3.563, size = 18076, normalized size = 564.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c)^p)^(1/p),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c)^p)^(1/p),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c)^p)^(1/p), x)

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Fricas [A]  time = 1.09594, size = 59, normalized size = 1.84 \begin{align*} -\frac{b^{\left (\frac{1}{p}\right )} \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((b*tan(d*x+c)^p)^(1/p),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac{1}{p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c)**p)**(1/p),x)

[Out]

Integral((b*tan(c + d*x)**p)**(1/p), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c)^p)^(1/p),x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c)^p)^(1/p), x)