3.3.16 \(\int \sec ^4(c+b x) \sin (a+b x) \, dx\) [216]
Optimal. Leaf size=67 \[ \frac {\cos (a-c) \sec ^3(c+b x)}{3 b}+\frac {\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{2 b}+\frac {\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b} \]
[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
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {4676, 2686, 30,
3853, 3855} \begin {gather*} \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} \end {gather*}
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 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*} \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) \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 {\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*}
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Mathematica [A]
time = 0.48, size = 64, normalized size = 0.96 \begin {gather*} \frac {12 \tanh ^{-1}\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} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. \(1781\) vs.
\(2(61)=122\).
time = 2.02, size = 1782, normalized size = 26.60
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {-3 \,{\mathrm e}^{i \left (5 b x +7 a +4 c \right )}+3 \,{\mathrm e}^{i \left (5 b x +5 a +6 c \right )}+8 \,{\mathrm e}^{i \left (3 b x +7 a +2 c \right )}+8 \,{\mathrm e}^{i \left (3 b x +5 a +4 c \right )}+3 \,{\mathrm e}^{i \left (b x +7 a \right )}-3 \,{\mathrm e}^{i \left (b x +5 a +2 c \right )}}{6 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{2 b}\) |
\(191\) |
| default |
\(\text {Expression too large to display}\) |
\(1782\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(b*x+c)^4*sin(b*x+a),x,method=_RETURNVERBOSE)
[Out]
1/b*(4*(-1/4*(sin(a)*cos(c)-cos(a)*sin(c))*(cos(a)^2*cos(c)^2+2*cos(a)*cos(c)*sin(a)*sin(c)+sin(a)^2*sin(c)^2)
/(cos(a)^4*cos(c)^4+2*sin(a)^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*sin
(c)^2*cos(a)^2*sin(a)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*s
in(a)^4)*tan(1/2*b*x+1/2*a)^5-1/4*(2*cos(a)^4*cos(c)^4-sin(a)^2*cos(a)^2*cos(c)^4+2*cos(c)^4*sin(a)^4+10*cos(a
)^3*cos(c)^3*sin(a)*sin(c)-10*sin(a)^3*cos(a)*sin(c)*cos(c)^3-cos(a)^4*sin(c)^2*cos(c)^2+28*cos(c)^2*sin(c)^2*
cos(a)^2*sin(a)^2-sin(a)^4*sin(c)^2*cos(c)^2-10*sin(a)*cos(a)^3*sin(c)^3*cos(c)+10*cos(c)*sin(c)^3*cos(a)*sin(
a)^3+2*sin(c)^4*cos(a)^4-sin(a)^2*cos(a)^2*sin(c)^4+2*sin(c)^4*sin(a)^4)/(cos(a)*cos(c)+sin(a)*sin(c))/(cos(a)
^4*cos(c)^4+2*sin(a)^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*sin(c)^2*co
s(a)^2*sin(a)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^4)
*tan(1/2*b*x+1/2*a)^4+1/6*(sin(a)*cos(c)-cos(a)*sin(c))*(6*cos(a)^4*cos(c)^4-7*sin(a)^2*cos(a)^2*cos(c)^4+2*co
s(c)^4*sin(a)^4+38*cos(a)^3*cos(c)^3*sin(a)*sin(c)-22*sin(a)^3*cos(a)*sin(c)*cos(c)^3-7*cos(a)^4*sin(c)^2*cos(
c)^2+76*cos(c)^2*sin(c)^2*cos(a)^2*sin(a)^2-7*sin(a)^4*sin(c)^2*cos(c)^2-22*sin(a)*cos(a)^3*sin(c)^3*cos(c)+38
*cos(c)*sin(c)^3*cos(a)*sin(a)^3+2*sin(c)^4*cos(a)^4-7*sin(a)^2*cos(a)^2*sin(c)^4+6*sin(c)^4*sin(a)^4)/(cos(a)
*cos(c)+sin(a)*sin(c))^2/(cos(a)^4*cos(c)^4+2*sin(a)^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2
*cos(c)^2+4*cos(c)^2*sin(c)^2*cos(a)^2*sin(a)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(
a)^2*sin(c)^4+sin(c)^4*sin(a)^4)*tan(1/2*b*x+1/2*a)^3-1/2*(4*cos(a)^2*cos(c)^3*sin(a)-cos(c)^3*sin(a)^3-4*cos(
a)^3*cos(c)^2*sin(c)+11*cos(c)^2*sin(c)*cos(a)*sin(a)^2-11*cos(c)*sin(c)^2*cos(a)^2*sin(a)+4*cos(c)*sin(c)^2*s
in(a)^3+sin(c)^3*cos(a)^3-4*sin(c)^3*cos(a)*sin(a)^2)/(cos(a)*cos(c)+sin(a)*sin(c))*(sin(a)*cos(c)-cos(a)*sin(
c))/(cos(a)^4*cos(c)^4+2*sin(a)^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*
sin(c)^2*cos(a)^2*sin(a)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^
4*sin(a)^4)*tan(1/2*b*x+1/2*a)^2-1/4*(3*cos(a)^2*cos(c)^2-2*cos(c)^2*sin(a)^2+10*cos(a)*cos(c)*sin(a)*sin(c)-2
*cos(a)^2*sin(c)^2+3*sin(a)^2*sin(c)^2)*(sin(a)*cos(c)-cos(a)*sin(c))/(cos(a)^4*cos(c)^4+2*sin(a)^2*cos(a)^2*c
os(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*sin(c)^2*cos(a)^2*sin(a)^2+2*sin(a)^4*sin(c)
^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^4)*tan(1/2*b*x+1/2*a)-1/12*(2*cos(c
)^3*cos(a)^3-cos(c)^3*cos(a)*sin(a)^2+8*cos(c)^2*sin(c)*cos(a)^2*sin(a)-cos(c)^2*sin(c)*sin(a)^3-cos(c)*sin(c)
^2*cos(a)^3+8*cos(c)*sin(c)^2*cos(a)*sin(a)^2-sin(c)^3*cos(a)^2*sin(a)+2*sin(c)^3*sin(a)^3)/(cos(a)^4*cos(c)^4
+2*sin(a)^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*sin(c)^2*cos(a)^2*sin(
a)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^4))/(cos(a)*c
os(c)*tan(1/2*b*x+1/2*a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-2*tan(1/2*b*x
+1/2*a)*sin(a)*cos(c)-cos(a)*cos(c)-sin(a)*sin(c))^3-(sin(a)*cos(c)-cos(a)*sin(c))/(cos(a)^4*cos(c)^4+2*sin(a)
^2*cos(a)^2*cos(c)^4+cos(c)^4*sin(a)^4+2*cos(a)^4*sin(c)^2*cos(c)^2+4*cos(c)^2*sin(c)^2*cos(a)^2*sin(a)^2+2*si
n(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^4)/(-cos(c)^2*sin(a)^2
-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)^2*sin(c)^2)^(1/2)*arctan(1/2*(2*(cos(a)*cos(c)+sin(a)*sin(c))*tan(
1/2*b*x+1/2*a)-2*sin(a)*cos(c)+2*cos(a)*sin(c))/(-cos(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)
^2*sin(c)^2)^(1/2)))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1424 vs.
\(2 (61) = 122\).
time = 0.59, size = 1424, normalized size = 21.25 \begin {gather*} \text {Too large to display} \end {gather*}
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))
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Fricas [A]
time = 2.17, size = 94, normalized size = 1.40 \begin {gather*} -\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 {gather*}
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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (61) = 122\).
time = 0.44, size = 495, normalized size = 7.39 \begin {gather*} \frac {\frac {3 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {3 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{5} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{5} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{5} \tan \left (\frac {1}{2} \, a\right ) + 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{5} \tan \left (\frac {1}{2} \, c\right ) - 12 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right ) - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{2} - 1\right )}}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, b} \end {gather*}
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
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(a + b*x)/cos(c + b*x)^4,x)
[Out]
\text{Hanged}
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