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

Optimal. Leaf size=62 \[ \frac{2 \tan (c+d x) \, _2F_1\left (1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right )}{d (2-p) \sqrt{b \tan ^p(c+d x)}} \]

[Out]

(2*Hypergeometric2F1[1, (2 - p)/4, (6 - p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(2 - p)*Sqrt[b*Tan[c + d*x]^p]
)

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Rubi [A]  time = 0.0485733, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac{2 \tan (c+d x) \, _2F_1\left (1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right )}{d (2-p) \sqrt{b \tan ^p(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*Tan[c + d*x]^p],x]

[Out]

(2*Hypergeometric2F1[1, (2 - p)/4, (6 - p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(2 - p)*Sqrt[b*Tan[c + d*x]^p]
)

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 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{b \tan ^p(c+d x)}} \, dx &=\frac{\tan ^{\frac{p}{2}}(c+d x) \int \tan ^{-\frac{p}{2}}(c+d x) \, dx}{\sqrt{b \tan ^p(c+d x)}}\\ &=\frac{\tan ^{\frac{p}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{x^{-p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt{b \tan ^p(c+d x)}}\\ &=\frac{2 \, _2F_1\left (1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d (2-p) \sqrt{b \tan ^p(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0485459, size = 60, normalized size = 0.97 \[ -\frac{2 \tan (c+d x) \, _2F_1\left (1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right )}{d (p-2) \sqrt{b \tan ^p(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*Tan[c + d*x]^p],x]

[Out]

(-2*Hypergeometric2F1[1, (2 - p)/4, (6 - p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(-2 + p)*Sqrt[b*Tan[c + d*x]^
p])

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Maple [F]  time = 0.126, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{p}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*tan(d*x+c)^p)^(1/2),x)

[Out]

int(1/(b*tan(d*x+c)^p)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*tan(d*x+c)^p)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*tan(d*x+c)^p)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*tan(d*x+c)**p)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*tan(d*x+c)^p)^(1/2),x, algorithm="giac")

[Out]

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