3.108 \(\int \sin (a+b x) \tan ^4(a+b x) \, dx\)

Optimal. Leaf size=38 \[ -\frac{\cos (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}-\frac{2 \sec (a+b x)}{b} \]

[Out]

-(Cos[a + b*x]/b) - (2*Sec[a + b*x])/b + Sec[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0252488, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2590, 270} \[ -\frac{\cos (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}-\frac{2 \sec (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]*Tan[a + b*x]^4,x]

[Out]

-(Cos[a + b*x]/b) - (2*Sec[a + b*x])/b + Sec[a + b*x]^3/(3*b)

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sin (a+b x) \tan ^4(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}-\frac{2}{x^2}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos (a+b x)}{b}-\frac{2 \sec (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0245295, size = 38, normalized size = 1. \[ -\frac{\cos (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}-\frac{2 \sec (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[a + b*x]/b) - (2*Sec[a + b*x])/b + Sec[a + b*x]^3/(3*b)

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Maple [A]  time = 0.021, size = 70, normalized size = 1.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{3\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{\cos \left ( bx+a \right ) }}- \left ({\frac{8}{3}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \cos \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^4*sin(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 0.97019, size = 47, normalized size = 1.24 \begin{align*} -\frac{\frac{6 \, \cos \left (b x + a\right )^{2} - 1}{\cos \left (b x + a\right )^{3}} + 3 \, \cos \left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.63776, size = 90, normalized size = 2.37 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{4} + 6 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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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)**4*sin(b*x+a)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.17805, size = 135, normalized size = 3.55 \begin{align*} \frac{2 \,{\left (\frac{3}{\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1} - \frac{\frac{12 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{3 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 5}{{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}}\right )}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^5,x, algorithm="giac")

[Out]

2/3*(3/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1) - (12*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 3*(cos(b*x +
a) - 1)^2/(cos(b*x + a) + 1)^2 + 5)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^3)/b