3.290 \(\int \sec ^4(e+f x) (a+b \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]

[Out]

(a*Tan[e + f*x])/f + ((a + b)*Tan[e + f*x]^3)/(3*f)

________________________________________________________________________________________

Rubi [A]  time = 0.032005, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3191} \[ \frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]

[Out]

(a*Tan[e + f*x])/f + ((a + b)*Tan[e + f*x]^3)/(3*f)

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0741321, size = 41, normalized size = 1.37 \[ \frac{a \left (\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f}+\frac{b \tan ^3(e+f x)}{3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]

[Out]

(b*Tan[e + f*x]^3)/(3*f) + (a*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 46, normalized size = 1.5 \begin{align*}{\frac{1}{f} \left ( -a \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \tan \left ( fx+e \right ) +{\frac{b \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x)

[Out]

1/f*(-a*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)+1/3*b*sin(f*x+e)^3/cos(f*x+e)^3)

________________________________________________________________________________________

Maxima [A]  time = 0.979309, size = 36, normalized size = 1.2 \begin{align*} \frac{{\left (a + b\right )} \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

1/3*((a + b)*tan(f*x + e)^3 + 3*a*tan(f*x + e))/f

________________________________________________________________________________________

Fricas [A]  time = 1.87285, size = 97, normalized size = 3.23 \begin{align*} \frac{{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

1/3*((2*a - b)*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(f*cos(f*x + e)^3)

________________________________________________________________________________________

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(f*x+e)**4*(a+b*sin(f*x+e)**2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.16849, size = 51, normalized size = 1.7 \begin{align*} \frac{a \tan \left (f x + e\right )^{3} + b \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm="giac")

[Out]

1/3*(a*tan(f*x + e)^3 + b*tan(f*x + e)^3 + 3*a*tan(f*x + e))/f