\(\int \sec ^4(c+b x) \sin (a+b x) \, dx\) [216]

Optimal result

Integrand size = 15, antiderivative size = 67 \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\frac {\cos (a-c) \sec ^3(c+b x)}{3 b}+\frac {\text {arctanh}(\sin (c+b x)) \sin (a-c)}{2 b}+\frac {\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4676, 2686, 30, 3853, 3855} \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\frac {\sin (a-c) \text {arctanh}(\sin (b x+c))}{2 b}+\frac {\cos (a-c) \sec ^3(b x+c)}{3 b}+\frac {\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b} \]

[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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2686

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 3853

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4676

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]

Rubi steps \begin{align*} \text {integral}& = \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) \text {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 {\text {arctanh}(\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] (verified)

Time = 0.54 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\frac {12 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a-c)+\sec ^3(c+b x) (4 \cos (a-c)+3 \sin (a-c) \sin (2 (c+b x)))}{12 b} \]

[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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.45 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.85

method	result	size
risch	\(\frac {-3 \,{\mathrm e}^{i \left (5 x b +7 a +4 c \right )}+3 \,{\mathrm e}^{i \left (5 x b +5 a +6 c \right )}+8 \,{\mathrm e}^{i \left (3 x b +7 a +2 c \right )}+8 \,{\mathrm e}^{i \left (3 x b +5 a +4 c \right )}+3 \,{\mathrm e}^{i \left (x b +7 a \right )}-3 \,{\mathrm e}^{i \left (x b +5 a +2 c \right )}}{6 b \left ({\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}\)	\(191\)
default	\(\text {Expression too large to display}\)	\(1782\)

[In]

int(sec(b*x+c)^4*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/6/b/(exp(2*I*(b*x+a+c))+exp(2*I*a))^3*(-3*exp(I*(5*b*x+7*a+4*c))+3*exp(I*(5*b*x+5*a+6*c))+8*exp(I*(3*b*x+7*a
+2*c))+8*exp(I*(3*b*x+5*a+4*c))+3*exp(I*(b*x+7*a))-3*exp(I*(b*x+5*a+2*c)))+1/2*ln(exp(I*(b*x+a))+I*exp(I*(a-c)
))/b*sin(a-c)-1/2*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/b*sin(a-c)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=-\frac {3 \, \cos \left (b x + c\right )^{3} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 3 \, \cos \left (b x + c\right )^{3} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) + 6 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 4 \, \cos \left (-a + c\right )}{12 \, b \cos \left (b x + c\right )^{3}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\text {Timed out} \]

[In]

integrate(sec(b*x+c)**4*sin(b*x+a),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1424 vs. \(2 (61) = 122\).

Time = 0.44 (sec) , antiderivative size = 1424, normalized size of antiderivative = 21.25 \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (61) = 122\).

Time = 0.31 (sec) , antiderivative size = 495, normalized size of antiderivative = 7.39 \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \sec ^4(c+b x) \sin (a+b x) \, dx=\text {Hanged} \]

[In]

int(sin(a + b*x)/cos(c + b*x)^4,x)

[Out]

\text{Hanged}