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

Optimal. Leaf size=61 \[ \frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.027738, antiderivative size = 61, 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 = {3658, 3473, 3475} \[ \frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x]^2)^(3/2),x]

[Out]

(b*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[b*Tan[c + d*x]^2])/d + (b*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^2])/(2*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 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 ^2(c+d x)\right )^{3/2} \, dx &=\left (b \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=\frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}-\left (b \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=\frac{b \cot (c+d x) \log (\cos (c+d x)) \sqrt{b \tan ^2(c+d x)}}{d}+\frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.115158, size = 47, normalized size = 0.77 \[ \frac{\cot ^3(c+d x) \left (b \tan ^2(c+d x)\right )^{3/2} \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x]^2)^(3/2),x]

[Out]

(Cot[c + d*x]^3*(b*Tan[c + d*x]^2)^(3/2)*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)

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Maple [A]  time = 0.019, size = 48, normalized size = 0.8 \begin{align*} -{\frac{- \left ( \tan \left ( dx+c \right ) \right ) ^{2}+\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ( \tan \left ( dx+c \right ) \right ) ^{3}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49755, size = 46, normalized size = 0.75 \begin{align*} \frac{b^{\frac{3}{2}} \tan \left (d x + c\right )^{2} - b^{\frac{3}{2}} \log \left (\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)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.37052, size = 135, normalized size = 2.21 \begin{align*} \frac{{\left (b \tan \left (d x + c\right )^{2} + b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + b\right )} \sqrt{b \tan \left (d x + c\right )^{2}}}{2 \, d \tan \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.51155, size = 55, normalized size = 0.9 \begin{align*} \frac{1}{2} \, b^{\frac{3}{2}}{\left (\frac{\tan \left (d x + c\right )^{2}}{d} - \frac{\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d}\right )} \mathrm{sgn}\left (\tan \left (d x + c\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^(3/2)*(tan(d*x + c)^2/d - log(tan(d*x + c)^2 + 1)/d)*sgn(tan(d*x + c))