3.1.28 \(\int (b \tan ^4(e+f x))^p \, dx\) [28]

Optimal. Leaf size=59 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (1+4 p);\frac {1}{2} (3+4 p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^4(e+f x)\right )^p}{f (1+4 p)} \]

[Out]

hypergeom([1, 1/2+2*p],[3/2+2*p],-tan(f*x+e)^2)*tan(f*x+e)*(b*tan(f*x+e)^4)^p/f/(1+4*p)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3739, 3557, 371} \begin {gather*} \frac {\tan (e+f x) \left (b \tan ^4(e+f x)\right )^p \, _2F_1\left (1,\frac {1}{2} (4 p+1);\frac {1}{2} (4 p+3);-\tan ^2(e+f x)\right )}{f (4 p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Tan[e + f*x]^4)^p,x]

[Out]

(Hypergeometric2F1[1, (1 + 4*p)/2, (3 + 4*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^4)^p)/(f*(1 + 4*
p))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

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 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, 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
]])

Rubi steps

\begin {align*} \int \left (b \tan ^4(e+f x)\right )^p \, dx &=\left (\tan ^{-4 p}(e+f x) \left (b \tan ^4(e+f x)\right )^p\right ) \int \tan ^{4 p}(e+f x) \, dx\\ &=\frac {\left (\tan ^{-4 p}(e+f x) \left (b \tan ^4(e+f x)\right )^p\right ) \text {Subst}\left (\int \frac {x^{4 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+4 p);\frac {1}{2} (3+4 p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^4(e+f x)\right )^p}{f (1+4 p)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.90 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2}+2 p;\frac {3}{2}+2 p;-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^4(e+f x)\right )^p}{f+4 f p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[e + f*x]^4)^p,x]

[Out]

(Hypergeometric2F1[1, 1/2 + 2*p, 3/2 + 2*p, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^4)^p)/(f + 4*f*p)

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (b \left (\tan ^{4}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(f*x+e)^4)^p,x)

[Out]

int((b*tan(f*x+e)^4)^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(f*x+e)^4)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^4)^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(f*x+e)^4)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^4)^p, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \tan ^{4}{\left (e + f x \right )}\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(f*x+e)**4)**p,x)

[Out]

Integral((b*tan(e + f*x)**4)**p, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(f*x+e)^4)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^4)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^4\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(e + f*x)^4)^p,x)

[Out]

int((b*tan(e + f*x)^4)^p, x)

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