3.113 \(\int \sec ^4(a+b x) \tan ^5(a+b x) \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^8(a+b x)}{8 b}+\frac{\tan ^6(a+b x)}{6 b} \]

[Out]

Tan[a + b*x]^6/(6*b) + Tan[a + b*x]^8/(8*b)

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Rubi [A]  time = 0.0311392, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 14} \[ \frac{\tan ^8(a+b x)}{8 b}+\frac{\tan ^6(a+b x)}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^4*Tan[a + b*x]^5,x]

[Out]

Tan[a + b*x]^6/(6*b) + Tan[a + b*x]^8/(8*b)

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sec ^4(a+b x) \tan ^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\tan ^6(a+b x)}{6 b}+\frac{\tan ^8(a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0552115, size = 38, normalized size = 1.23 \[ \frac{3 \sec ^8(a+b x)-8 \sec ^6(a+b x)+6 \sec ^4(a+b x)}{24 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^4*Tan[a + b*x]^5,x]

[Out]

(6*Sec[a + b*x]^4 - 8*Sec[a + b*x]^6 + 3*Sec[a + b*x]^8)/(24*b)

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Maple [A]  time = 0.023, size = 42, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{24\, \left ( \cos \left ( bx+a \right ) \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^9*sin(b*x+a)^5,x)

[Out]

1/b*(1/8*sin(b*x+a)^6/cos(b*x+a)^8+1/24*sin(b*x+a)^6/cos(b*x+a)^6)

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Maxima [B]  time = 0.988295, size = 93, normalized size = 3. \begin{align*} \frac{6 \, \sin \left (b x + a\right )^{4} - 4 \, \sin \left (b x + a\right )^{2} + 1}{24 \,{\left (\sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6} + 6 \, \sin \left (b x + a\right )^{4} - 4 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^9*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

1/24*(6*sin(b*x + a)^4 - 4*sin(b*x + a)^2 + 1)/((sin(b*x + a)^8 - 4*sin(b*x + a)^6 + 6*sin(b*x + a)^4 - 4*sin(
b*x + a)^2 + 1)*b)

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Fricas [A]  time = 1.53711, size = 90, normalized size = 2.9 \begin{align*} \frac{6 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} + 3}{24 \, b \cos \left (b x + a\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^9*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

1/24*(6*cos(b*x + a)^4 - 8*cos(b*x + a)^2 + 3)/(b*cos(b*x + a)^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**9*sin(b*x+a)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.19924, size = 126, normalized size = 4.06 \begin{align*} -\frac{32 \,{\left (\frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}}\right )}}{3 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^9*sin(b*x+a)^5,x, algorithm="giac")

[Out]

-32/3*((cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - (cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + (cos(b*x + a) -
1)^5/(cos(b*x + a) + 1)^5)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^8)