3.37 \(\int (b \tan ^4(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=110 \[ \frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}-b x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}+\frac{b \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d} \]

[Out]

(b*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - b*x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4] - (b*Tan[c + d*x]*Sqrt[b
*Tan[c + d*x]^4])/(3*d) + (b*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^4])/(5*d)

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Rubi [A]  time = 0.0427885, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ \frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}-b x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}+\frac{b \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x]^4)^(3/2),x]

[Out]

(b*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - b*x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4] - (b*Tan[c + d*x]*Sqrt[b
*Tan[c + d*x]^4])/(3*d) + (b*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^4])/(5*d)

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
]])

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 8

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

Rubi steps

\begin{align*} \int \left (b \tan ^4(c+d x)\right )^{3/2} \, dx &=\left (b \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int \tan ^6(c+d x) \, dx\\ &=\frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-\left (b \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int \tan ^4(c+d x) \, dx\\ &=-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}+\frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}+\left (b \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac{b \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}+\frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-\left (b \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac{b \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-b x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}+\frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.743299, size = 66, normalized size = 0.6 \[ \frac{\cot (c+d x) \left (b \tan ^4(c+d x)\right )^{3/2} \left (15 \cot ^4(c+d x)-5 \cot ^2(c+d x)-15 \tan ^{-1}(\tan (c+d x)) \cot ^5(c+d x)+3\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x]^4)^(3/2),x]

[Out]

(Cot[c + d*x]*(3 - 5*Cot[c + d*x]^2 + 15*Cot[c + d*x]^4 - 15*ArcTan[Tan[c + d*x]]*Cot[c + d*x]^5)*(b*Tan[c + d
*x]^4)^(3/2))/(15*d)

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Maple [A]  time = 0.015, size = 64, normalized size = 0.6 \begin{align*} -{\frac{-3\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}+5\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}+15\,\arctan \left ( \tan \left ( dx+c \right ) \right ) -15\,\tan \left ( dx+c \right ) }{15\,d \left ( \tan \left ( dx+c \right ) \right ) ^{6}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{4} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c)^4)^(3/2),x)

[Out]

-1/15/d*(b*tan(d*x+c)^4)^(3/2)*(-3*tan(d*x+c)^5+5*tan(d*x+c)^3+15*arctan(tan(d*x+c))-15*tan(d*x+c))/tan(d*x+c)
^6

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Maxima [A]  time = 1.41117, size = 72, normalized size = 0.65 \begin{align*} \frac{3 \, b^{\frac{3}{2}} \tan \left (d x + c\right )^{5} - 5 \, b^{\frac{3}{2}} \tan \left (d x + c\right )^{3} - 15 \,{\left (d x + c\right )} b^{\frac{3}{2}} + 15 \, b^{\frac{3}{2}} \tan \left (d x + c\right )}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(3/2),x, algorithm="maxima")

[Out]

1/15*(3*b^(3/2)*tan(d*x + c)^5 - 5*b^(3/2)*tan(d*x + c)^3 - 15*(d*x + c)*b^(3/2) + 15*b^(3/2)*tan(d*x + c))/d

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Fricas [A]  time = 1.41854, size = 163, normalized size = 1.48 \begin{align*} \frac{{\left (3 \, b \tan \left (d x + c\right )^{5} - 5 \, b \tan \left (d x + c\right )^{3} - 15 \, b d x + 15 \, b \tan \left (d x + c\right )\right )} \sqrt{b \tan \left (d x + c\right )^{4}}}{15 \, d \tan \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(3/2),x, algorithm="fricas")

[Out]

1/15*(3*b*tan(d*x + c)^5 - 5*b*tan(d*x + c)^3 - 15*b*d*x + 15*b*tan(d*x + c))*sqrt(b*tan(d*x + c)^4)/(d*tan(d*
x + c)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)**4*b)**(3/2),x)

[Out]

Integral((b*tan(c + d*x)**4)**(3/2), x)

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Giac [B]  time = 6.32711, size = 1339, normalized size = 12.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^(3/2),x, algorithm="giac")

[Out]

1/60*(15*pi - 60*d*x*tan(d*x)^5*tan(c)^5 - 15*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) -
2*tan(c))*tan(d*x)^5*tan(c)^5 - 15*pi*tan(d*x)^5*tan(c)^5 + 30*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)
))*tan(d*x)^5*tan(c)^5 + 30*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^5*tan(c)^5 + 300*d*x*ta
n(d*x)^4*tan(c)^4 + 75*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*ta
n(c)^4 + 75*pi*tan(d*x)^4*tan(c)^4 - 150*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)^4*tan(c)^4
- 150*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 - 60*tan(d*x)^5*tan(c)^4 - 60*tan
(d*x)^4*tan(c)^5 - 600*d*x*tan(d*x)^3*tan(c)^3 - 150*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(
d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 20*tan(d*x)^5*tan(c)^2 - 150*pi*tan(d*x)^3*tan(c)^3 + 300*arctan((tan(d
*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)^3*tan(c)^3 + 300*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1
))*tan(d*x)^3*tan(c)^3 + 300*tan(d*x)^4*tan(c)^3 + 300*tan(d*x)^3*tan(c)^4 + 20*tan(d*x)^2*tan(c)^5 + 600*d*x*
tan(d*x)^2*tan(c)^2 + 150*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)^2
*tan(c)^2 - 12*tan(d*x)^5 - 100*tan(d*x)^4*tan(c) + 150*pi*tan(d*x)^2*tan(c)^2 - 300*arctan((tan(d*x)*tan(c) -
1)/(tan(d*x) + tan(c)))*tan(d*x)^2*tan(c)^2 - 300*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^
2*tan(c)^2 - 600*tan(d*x)^3*tan(c)^2 - 600*tan(d*x)^2*tan(c)^3 - 100*tan(d*x)*tan(c)^4 - 12*tan(c)^5 - 300*d*x
*tan(d*x)*tan(c) - 75*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c
) + 20*tan(d*x)^3 - 75*pi*tan(d*x)*tan(c) + 150*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)*tan
(c) + 150*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) + 300*tan(d*x)^2*tan(c) + 300*tan(
d*x)*tan(c)^2 + 20*tan(c)^3 + 60*d*x + 15*pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*ta
n(c)) - 30*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c))) - 30*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c)
- 1)) - 60*tan(d*x) - 60*tan(c))*b^(3/2)/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d*x)^4*tan(c)^4 + 10*d*tan(d*x)^3*ta
n(c)^3 - 10*d*tan(d*x)^2*tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)