3.31 \(\int \sin ^3(e+f x) (a+b \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=48 \[ \frac {(a-b) \cos ^3(e+f x)}{3 f}-\frac {(a-2 b) \cos (e+f x)}{f}+\frac {b \sec (e+f x)}{f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3664, 448} \[ \frac {(a-b) \cos ^3(e+f x)}{3 f}-\frac {(a-2 b) \cos (e+f x)}{f}+\frac {b \sec (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]

[Out]

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 72, normalized size = 1.50 \[ -\frac {3 a \cos (e+f x)}{4 f}+\frac {a \cos (3 (e+f x))}{12 f}+\frac {7 b \cos (e+f x)}{4 f}-\frac {b \cos (3 (e+f x))}{12 f}+\frac {b \sec (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]

[Out]

(-3*a*Cos[e + f*x])/(4*f) + (7*b*Cos[e + f*x])/(4*f) + (a*Cos[3*(e + f*x)])/(12*f) - (b*Cos[3*(e + f*x)])/(12*
f) + (b*Sec[e + f*x])/f

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 46, normalized size = 0.96 \[ \frac {{\left (a - b\right )} \cos \left (f x + e\right )^{4} - 3 \, {\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \, f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.89, size = 76, normalized size = 1.58 \[ \frac {b}{f \cos \left (f x + e\right )} + \frac {a f^{5} \cos \left (f x + e\right )^{3} - b f^{5} \cos \left (f x + e\right )^{3} - 3 \, a f^{5} \cos \left (f x + e\right ) + 6 \, b f^{5} \cos \left (f x + e\right )}{3 \, f^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm="giac")

[Out]

b/(f*cos(f*x + e)) + 1/3*(a*f^5*cos(f*x + e)^3 - b*f^5*cos(f*x + e)^3 - 3*a*f^5*cos(f*x + e) + 6*b*f^5*cos(f*x
+ e))/f^6

________________________________________________________________________________________

maple [A]  time = 0.60, size = 72, normalized size = 1.50 \[ \frac {-\frac {a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+b \left (\frac {\sin ^{6}\left (f x +e \right )}{\cos \left (f x +e \right )}+\left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^3*(a+b*tan(f*x+e)^2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 44, normalized size = 0.92 \[ \frac {{\left (a - b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a - 2 \, b\right )} \cos \left (f x + e\right ) + \frac {3 \, b}{\cos \left (f x + e\right )}}{3 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 12.01, size = 68, normalized size = 1.42 \[ -\frac {\frac {3\,a}{8}-\frac {15\,b}{8}+\frac {a\,\cos \left (2\,e+2\,f\,x\right )}{3}-\frac {a\,\cos \left (4\,e+4\,f\,x\right )}{24}-\frac {5\,b\,\cos \left (2\,e+2\,f\,x\right )}{6}+\frac {b\,\cos \left (4\,e+4\,f\,x\right )}{24}}{f\,\cos \left (e+f\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^3*(a + b*tan(e + f*x)^2),x)

[Out]

-((3*a)/8 - (15*b)/8 + (a*cos(2*e + 2*f*x))/3 - (a*cos(4*e + 4*f*x))/24 - (5*b*cos(2*e + 2*f*x))/6 + (b*cos(4*
e + 4*f*x))/24)/(f*cos(e + f*x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right ) \sin ^{3}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**3*(a+b*tan(f*x+e)**2),x)

[Out]

Integral((a + b*tan(e + f*x)**2)*sin(e + f*x)**3, x)

________________________________________________________________________________________