3.38 \(\int \sqrt {b \tan ^4(c+d x)} \, dx\)

Optimal. Leaf size=50 \[ \frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \]

[Out]

cot(d*x+c)*(b*tan(d*x+c)^4)^(1/2)/d-x*cot(d*x+c)^2*(b*tan(d*x+c)^4)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {3658, 3473, 8} \[ \frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Tan[c + d*x]^4],x]

[Out]

(Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3658

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 \sqrt {b \tan ^4(c+d x)} \, dx &=\left (\cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\left (\cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 41, normalized size = 0.82 \[ -\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)} \left (\tan ^{-1}(\tan (c+d x)) \cot (c+d x)-1\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Tan[c + d*x]^4],x]

[Out]

-((Cot[c + d*x]*(-1 + ArcTan[Tan[c + d*x]]*Cot[c + d*x])*Sqrt[b*Tan[c + d*x]^4])/d)

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fricas [A]  time = 1.03, size = 37, normalized size = 0.74 \[ -\frac {\sqrt {b \tan \left (d x + c\right )^{4}} {\left (d x - \tan \left (d x + c\right )\right )}}{d \tan \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(b*tan(d*x + c)^4)*(d*x - tan(d*x + c))/(d*tan(d*x + c)^2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)sqrt(b)*(-4*d*x*tan(c)*tan(d*x)+4*d*x-pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^
2-2*tan(c)-2*tan(d*x))*tan(c)*tan(d*x)+pi*sign(2*tan(c)^2*tan(d*x)+2*tan(c)*tan(d*x)^2-2*tan(c)-2*tan(d*x))-pi
*tan(c)*tan(d*x)+pi+2*atan((tan(c)*tan(d*x)-1)/(tan(c)+tan(d*x)))*tan(c)*tan(d*x)-2*atan((tan(c)*tan(d*x)-1)/(
tan(c)+tan(d*x)))+2*atan((tan(c)+tan(d*x))/(tan(c)*tan(d*x)-1))*tan(c)*tan(d*x)-2*atan((tan(c)+tan(d*x))/(tan(
c)*tan(d*x)-1))-4*tan(c)-4*tan(d*x))/(4*d*tan(c)*tan(d*x)-4*d)

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maple [A]  time = 0.10, size = 42, normalized size = 0.84 \[ -\frac {\sqrt {b \left (\tan ^{4}\left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d \tan \left (d x +c \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c)^4)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.64, size = 26, normalized size = 0.52 \[ -\frac {{\left (d x + c\right )} \sqrt {b} - \sqrt {b} \tan \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(1/2),x, algorithm="maxima")

[Out]

-((d*x + c)*sqrt(b) - sqrt(b)*tan(d*x + c))/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(c + d*x)^4)^(1/2),x)

[Out]

int((b*tan(c + d*x)^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan ^{4}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)**4*b)**(1/2),x)

[Out]

Integral(sqrt(b*tan(c + d*x)**4), x)

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