3.216 \(\int \sec ^4(c+b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=67 \[ \frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec ^3(b x+c)}{3 b}+\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b} \]
[Out]
(Cos[a - c]*Sec[c + b*x]^3)/(3*b) + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/(2*b) + (Sec[c + b*x]*Sin[a - c]*Tan[c
+ b*x])/(2*b)
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Rubi [A] time = 0.0417556, antiderivative size = 67, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{4580, 2606, 30, 3768, 3770} \[ \frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec ^3(b x+c)}{3 b}+\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[Sec[c + b*x]^4*Sin[a + b*x],x]
[Out]
(Cos[a - c]*Sec[c + b*x]^3)/(3*b) + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/(2*b) + (Sec[c + b*x]*Sin[a - c]*Tan[c
+ b*x])/(2*b)
Rule 4580
Int[Sec[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Cos[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Sin[v - w],
Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 3768
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \sec ^4(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^3(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^3(c+b x) \, dx\\ &=\frac{\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b}+\frac{\cos (a-c) \operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+b x)\right )}{b}+\frac{1}{2} \sin (a-c) \int \sec (c+b x) \, dx\\ &=\frac{\cos (a-c) \sec ^3(c+b x)}{3 b}+\frac{\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{2 b}+\frac{\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.430755, size = 64, normalized size = 0.96 \[ \frac{\sec ^3(b x+c) (3 \sin (a-c) \sin (2 (b x+c))+4 \cos (a-c))+12 \sin (a-c) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{b x}{2}\right )+\sin (c)\right )}{12 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + b*x]^4*Sin[a + b*x],x]
[Out]
(12*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c] + Sec[c + b*x]^3*(4*Cos[a - c] + 3*Sin[a - c]*Sin[2*(c +
b*x)]))/(12*b)
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Maple [B] time = 2.318, size = 14825, normalized size = 221.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(b*x+c)^4*sin(b*x+a),x)
[Out]
result too large to display
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Maxima [B] time = 2.16094, size = 1922, normalized size = 28.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^4*sin(b*x+a),x, algorithm="maxima")
[Out]
-1/12*(2*(3*cos(5*b*x + 2*a + 4*c) - 3*cos(5*b*x + 6*c) - 8*cos(3*b*x + 2*a + 2*c) - 8*cos(3*b*x + 4*c) - 3*co
s(b*x + 2*a) + 3*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) + 6*(3*cos(4*b*x + a + 4*c) + 3*cos(2*b*x + a + 2*c) + c
os(a))*cos(5*b*x + 2*a + 4*c) - 6*(3*cos(4*b*x + a + 4*c) + 3*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*x + 6*c)
- 6*(8*cos(3*b*x + 2*a + 2*c) + 8*cos(3*b*x + 4*c) + 3*cos(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(4*b*x + a + 4*c)
- 16*(3*cos(2*b*x + a + 2*c) + cos(a))*cos(3*b*x + 2*a + 2*c) - 16*(3*cos(2*b*x + a + 2*c) + cos(a))*cos(3*b*
x + 4*c) - 18*(cos(b*x + 2*a) - cos(b*x + 2*c))*cos(2*b*x + a + 2*c) - 6*cos(b*x + 2*a)*cos(a) + 6*cos(b*x + 2
*c)*cos(a) - 3*(cos(6*b*x + a + 6*c)^2*sin(-a + c) + 9*cos(4*b*x + a + 4*c)^2*sin(-a + c) + 9*cos(2*b*x + a +
2*c)^2*sin(-a + c) + 6*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(6*b*x + a + 6*c)^2*sin(-a + c) + 9*sin(4*
b*x + a + 4*c)^2*sin(-a + c) + 9*sin(2*b*x + a + 2*c)^2*sin(-a + c) + 6*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c
) + 2*(3*cos(4*b*x + a + 4*c)*sin(-a + c) + 3*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(6*b*x
+ a + 6*c) + 6*(3*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(4*b*x + a + 4*c) + 2*(3*sin(4*b*
x + a + 4*c)*sin(-a + c) + 3*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(6*b*x + a + 6*c) + 6*(
3*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(4*b*x + a + 4*c) + (cos(a)^2 + sin(a)^2)*sin(-a +
c))*log((cos(b*x + 2*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) +
sin(c)^2)/(cos(b*x + 2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c)
+ sin(c)^2)) + 2*(3*sin(5*b*x + 2*a + 4*c) - 3*sin(5*b*x + 6*c) - 8*sin(3*b*x + 2*a + 2*c) - 8*sin(3*b*x + 4*
c) - 3*sin(b*x + 2*a) + 3*sin(b*x + 2*c))*sin(6*b*x + a + 6*c) + 6*(3*sin(4*b*x + a + 4*c) + 3*sin(2*b*x + a +
2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 6*(3*sin(4*b*x + a + 4*c) + 3*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*
x + 6*c) - 6*(8*sin(3*b*x + 2*a + 2*c) + 8*sin(3*b*x + 4*c) + 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(4*b*x +
a + 4*c) - 16*(3*sin(2*b*x + a + 2*c) + sin(a))*sin(3*b*x + 2*a + 2*c) - 16*(3*sin(2*b*x + a + 2*c) + sin(a))
*sin(3*b*x + 4*c) - 18*(sin(b*x + 2*a) - sin(b*x + 2*c))*sin(2*b*x + a + 2*c) - 6*sin(b*x + 2*a)*sin(a) + 6*si
n(b*x + 2*c)*sin(a))/(b*cos(6*b*x + a + 6*c)^2 + 9*b*cos(4*b*x + a + 4*c)^2 + 9*b*cos(2*b*x + a + 2*c)^2 + 6*b
*cos(2*b*x + a + 2*c)*cos(a) + b*sin(6*b*x + a + 6*c)^2 + 9*b*sin(4*b*x + a + 4*c)^2 + 9*b*sin(2*b*x + a + 2*c
)^2 + 6*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b + 2*(3*b*cos(4*b*x + a + 4*c) + 3*b*cos(2*b*x
+ a + 2*c) + b*cos(a))*cos(6*b*x + a + 6*c) + 6*(3*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(4*b*x + a + 4*c) + 2
*(3*b*sin(4*b*x + a + 4*c) + 3*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(6*b*x + a + 6*c) + 6*(3*b*sin(2*b*x + a
+ 2*c) + b*sin(a))*sin(4*b*x + a + 4*c))
________________________________________________________________________________________
Fricas [A] time = 0.5273, size = 258, normalized size = 3.85 \begin{align*} -\frac{3 \, \cos \left (b x + c\right )^{3} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 3 \, \cos \left (b x + c\right )^{3} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) + 6 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 4 \, \cos \left (-a + c\right )}{12 \, b \cos \left (b x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^4*sin(b*x+a),x, algorithm="fricas")
[Out]
-1/12*(3*cos(b*x + c)^3*log(sin(b*x + c) + 1)*sin(-a + c) - 3*cos(b*x + c)^3*log(-sin(b*x + c) + 1)*sin(-a + c
) + 6*cos(b*x + c)*sin(b*x + c)*sin(-a + c) - 4*cos(-a + c))/(b*cos(b*x + c)^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+c)**4*sin(b*x+a),x)
[Out]
Timed out
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Giac [B] time = 1.20114, size = 668, normalized size = 9.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(b*x+c)^4*sin(b*x+a),x, algorithm="giac")
[Out]
1/3*(3*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2
*c) + 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - 3*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2
*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c) - 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + ta
n(1/2*a)^2 + tan(1/2*c)^2 + 1) + 2*(3*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^2*tan(1/2*c) - 3*tan(1/2*b*x + 1/2*c)^
5*tan(1/2*a)*tan(1/2*c)^2 - 3*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*b*x + 1/2*c)^5*tan(
1/2*a) + 3*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2 - 3*tan(1/2*b*x + 1/2*c)^5*tan(1/2*c) - 12*tan(1/2*b*x + 1/2*c)
^4*tan(1/2*a)*tan(1/2*c) + 3*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^2 - 3*tan(1/2*b*x + 1/2*c)^4 - 3*tan(1/2*b*x +
1/2*c)*tan(1/2*a)^2*tan(1/2*c) + 3*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 -
3*tan(1/2*b*x + 1/2*c)*tan(1/2*a) + tan(1/2*a)^2 + 3*tan(1/2*b*x + 1/2*c)*tan(1/2*c) - 4*tan(1/2*a)*tan(1/2*c)
+ tan(1/2*c)^2 - 1)/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*(tan(1/2*b*x + 1/2*c)^2 -
1)^3))/b