3.3.18 \(\int \sec ^6(c+b x) \sin (a+b x) \, dx\) [218]

Optimal. Leaf size=94 \[ \frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{8 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b} \]

[Out]

1/5*cos(a-c)*sec(b*x+c)^5/b+3/8*arctanh(sin(b*x+c))*sin(a-c)/b+3/8*sec(b*x+c)*sin(a-c)*tan(b*x+c)/b+1/4*sec(b*
x+c)^3*sin(a-c)*tan(b*x+c)/b

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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{8 b}+\frac {\cos (a-c) \sec ^5(b x+c)}{5 b}+\frac {\sin (a-c) \tan (b x+c) \sec ^3(b x+c)}{4 b}+\frac {3 \sin (a-c) \tan (b x+c) \sec (b x+c)}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + b*x]^6*Sin[a + b*x],x]

[Out]

(Cos[a - c]*Sec[c + b*x]^5)/(5*b) + (3*ArcTanh[Sin[c + b*x]]*Sin[a - c])/(8*b) + (3*Sec[c + b*x]*Sin[a - c]*Ta
n[c + b*x])/(8*b) + (Sec[c + b*x]^3*Sin[a - c]*Tan[c + b*x])/(4*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 ^6(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^5(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^5(c+b x) \, dx\\ &=\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac {\cos (a-c) \text {Subst}\left (\int x^4 \, dx,x,\sec (c+b x)\right )}{b}+\frac {1}{4} (3 \sin (a-c)) \int \sec ^3(c+b x) \, dx\\ &=\frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac {1}{8} (3 \sin (a-c)) \int \sec (c+b x) \, dx\\ &=\frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{8 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}\\ \end {align*}

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Mathematica [A]
time = 1.05, size = 78, normalized size = 0.83 \begin {gather*} \frac {480 \tanh ^{-1}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a-c)+2 \sec ^5(c+b x) (64 \cos (a-c)+5 \sin (a-c) (14 \sin (2 (c+b x))+3 \sin (4 (c+b x))))}{640 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + b*x]^6*Sin[a + b*x],x]

[Out]

(480*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c] + 2*Sec[c + b*x]^5*(64*Cos[a - c] + 5*Sin[a - c]*(14*Sin
[2*(c + b*x)] + 3*Sin[4*(c + b*x)])))/(640*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6741\) vs. \(2(86)=172\).
time = 6.90, size = 6742, normalized size = 71.72

method	result	size
risch	\(\frac {-15 \,{\mathrm e}^{i \left (9 b x +11 a +8 c \right )}+15 \,{\mathrm e}^{i \left (9 b x +9 a +10 c \right )}-70 \,{\mathrm e}^{i \left (7 b x +11 a +6 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +9 a +8 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +11 a +4 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +9 a +6 c \right )}+70 \,{\mathrm e}^{i \left (3 b x +11 a +2 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +9 a +4 c \right )}+15 \,{\mathrm e}^{i \left (b x +11 a \right )}-15 \,{\mathrm e}^{i \left (b x +9 a +2 c \right )}}{40 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{8 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{8 b}\)	\(259\)
default	\(\text {Expression too large to display}\)	\(6742\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+c)^6*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3096 vs. \(2 (86) = 172\).
time = 0.69, size = 3096, normalized size = 32.94 \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)^6*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/80*(2*(15*cos(9*b*x + 2*a + 8*c) - 15*cos(9*b*x + 10*c) + 70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) -
128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*cos(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(
b*x + 2*a) + 15*cos(b*x + 2*c))*cos(10*b*x + a + 10*c) + 30*(5*cos(8*b*x + a + 8*c) + 10*cos(6*b*x + a + 6*c)
+ 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 2*a + 8*c) - 30*(5*cos(8*b*x + a + 8*
c) + 10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 10*c) +
10*(70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) - 128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*c
os(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(8*b*x + a + 8*c) + 14
0*(10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(7*b*x + 2*a + 6*c)
- 140*(10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(7*b*x + 8*c)
- 20*(128*cos(5*b*x + 2*a + 4*c) + 128*cos(5*b*x + 6*c) + 70*cos(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15
*cos(b*x + 2*a) - 15*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) - 256*(10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2
*c) + cos(a))*cos(5*b*x + 2*a + 4*c) - 256*(10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b
*x + 6*c) - 100*(14*cos(3*b*x + 2*a + 2*c) - 14*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) - 140*(5*cos(2*b*x + a + 2*c) + cos(a))*cos(3*b*x + 2*a + 2*c) + 140*(5*cos(2*b*x + a + 2*c) +
cos(a))*cos(3*b*x + 4*c) - 150*(cos(b*x + 2*a) - cos(b*x + 2*c))*cos(2*b*x + a + 2*c) - 30*cos(b*x + 2*a)*cos(
a) + 30*cos(b*x + 2*c)*cos(a) - 15*(cos(10*b*x + a + 10*c)^2*sin(-a + c) + 25*cos(8*b*x + a + 8*c)^2*sin(-a +
c) + 100*cos(6*b*x + a + 6*c)^2*sin(-a + c) + 100*cos(4*b*x + a + 4*c)^2*sin(-a + c) + 25*cos(2*b*x + a + 2*c)
^2*sin(-a + c) + 10*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(10*b*x + a + 10*c)^2*sin(-a + c) + 25*sin(8*
b*x + a + 8*c)^2*sin(-a + c) + 100*sin(6*b*x + a + 6*c)^2*sin(-a + c) + 100*sin(4*b*x + a + 4*c)^2*sin(-a + c)
+ 25*sin(2*b*x + a + 2*c)^2*sin(-a + c) + 10*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + 2*(5*cos(8*b*x + a + 8
*c)*sin(-a + c) + 10*cos(6*b*x + a + 6*c)*sin(-a + c) + 10*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a
+ 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(10*b*x + a + 10*c) + 10*(10*cos(6*b*x + a + 6*c)*sin(-a + c) + 10
*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(8*b*x + a + 8
*c) + 20*(10*cos(4*b*x + a + 4*c)*sin(-a + c) + 5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(6
*b*x + a + 6*c) + 20*(5*cos(2*b*x + a + 2*c)*sin(-a + c) + cos(a)*sin(-a + c))*cos(4*b*x + a + 4*c) + 2*(5*sin
(8*b*x + a + 8*c)*sin(-a + c) + 10*sin(6*b*x + a + 6*c)*sin(-a + c) + 10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*
sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin(10*b*x + a + 10*c) + 10*(10*sin(6*b*x + a + 6*c)*si
n(-a + c) + 10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(-a + c))*sin
(8*b*x + a + 8*c) + 20*(10*sin(4*b*x + a + 4*c)*sin(-a + c) + 5*sin(2*b*x + a + 2*c)*sin(-a + c) + sin(a)*sin(
-a + c))*sin(6*b*x + a + 6*c) + 20*(5*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*(15*sin(9*b*x + 2*a + 8*c) - 15*sin(9*b*x + 10*c) + 70*s
in(7*b*x + 2*a + 6*c) - 70*sin(7*b*x + 8*c) - 128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x
+ 2*a + 2*c) + 70*sin(3*b*x + 4*c) - 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(10*b*x + a + 10*c) + 30*(5*si
n(8*b*x + a + 8*c) + 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(
9*b*x + 2*a + 8*c) - 30*(5*sin(8*b*x + a + 8*c) + 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*
b*x + a + 2*c) + sin(a))*sin(9*b*x + 10*c) + 10*(70*sin(7*b*x + 2*a + 6*c) - 70*sin(7*b*x + 8*c) - 128*sin(5*b
*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x + 2*a + 2*c) + 70*sin(3*b*x + 4*c) - 15*sin(b*x + 2*a) +
15*sin(b*x + 2*c))*sin(8*b*x + a + 8*c) + 140*(10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*
x + a + 2*c) + sin(a))*sin(7*b*x + 2*a + 6*c) - 140*(10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) + 5*sin
(2*b*x + a + 2*c) + sin(a))*sin(7*b*x + 8*c) - 20*(128*sin(5*b*x + 2*a + 4*c) + 128*sin(5*b*x + 6*c) + 70*sin(
3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4*c) + 15*sin(b*x + 2*a) - 15*sin(b*x + 2*c))*sin(6*b*x + a + 6*c) - 256*(
10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 256*(10*sin(4*b*x + a + 4*
c) + 5*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 6*c) - 100*(14*sin(3*b*x + 2*a + 2*c) - 14*sin(3*b*x + 4*c)
+ 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(4*b*...

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Fricas [A]
time = 2.95, size = 107, normalized size = 1.14 \begin {gather*} -\frac {15 \, \cos \left (b x + c\right )^{5} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 15 \, \cos \left (b x + c\right )^{5} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) + 10 \, {\left (3 \, \cos \left (b x + c\right )^{3} + 2 \, \cos \left (b x + c\right )\right )} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 16 \, \cos \left (-a + c\right )}{80 \, b \cos \left (b x + c\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+c)^6*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/80*(15*cos(b*x + c)^5*log(sin(b*x + c) + 1)*sin(-a + c) - 15*cos(b*x + c)^5*log(-sin(b*x + c) + 1)*sin(-a +
c) + 10*(3*cos(b*x + c)^3 + 2*cos(b*x + c))*sin(b*x + c)*sin(-a + c) - 16*cos(-a + c))/(b*cos(b*x + c)^5)

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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)**6*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. 756 vs. \(2 (86) = 172\).
time = 0.44, size = 756, normalized size = 8.04 \begin {gather*} \frac {\frac {15 \, {\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 {15 \, {\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 (25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right ) + 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right )^{2} - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, c\right ) - 80 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right ) + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, c\right ) - 40 \, \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} + 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} - 160 \, \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 ) + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - 25 \, \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 ) + 25 \, \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} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right ) - 16 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, c\right )^{2} - 4\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 )}^{5}}}{20 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+c)^6*sin(b*x+a),x, algorithm="giac")

[Out]

1/20*(15*(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) - 15*(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) + 2*(25*tan(1/2*b*x + 1/2*c)^9*tan(1/2*a)^2*tan(1/2*c) - 25*tan(1/2*b*x + 1/
2*c)^9*tan(1/2*a)*tan(1/2*c)^2 - 20*tan(1/2*b*x + 1/2*c)^8*tan(1/2*a)^2*tan(1/2*c)^2 + 25*tan(1/2*b*x + 1/2*c)
^9*tan(1/2*a) + 20*tan(1/2*b*x + 1/2*c)^8*tan(1/2*a)^2 - 25*tan(1/2*b*x + 1/2*c)^9*tan(1/2*c) - 80*tan(1/2*b*x
+ 1/2*c)^8*tan(1/2*a)*tan(1/2*c) - 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*a)^2*tan(1/2*c) + 20*tan(1/2*b*x + 1/2*c
)^8*tan(1/2*c)^2 + 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*a)*tan(1/2*c)^2 - 20*tan(1/2*b*x + 1/2*c)^8 - 10*tan(1/2*
b*x + 1/2*c)^7*tan(1/2*a) + 10*tan(1/2*b*x + 1/2*c)^7*tan(1/2*c) - 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2*tan(
1/2*c)^2 + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^2 - 160*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)*tan(1/2*c) + 10*tan(
1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c) + 40*tan(1/2*b*x + 1/2*c)^4*tan(1/2*c)^2 - 10*tan(1/2*b*x + 1/2*c)^
3*tan(1/2*a)*tan(1/2*c)^2 - 40*tan(1/2*b*x + 1/2*c)^4 + 10*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) - 10*tan(1/2*b*x
+ 1/2*c)^3*tan(1/2*c) - 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c) + 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*t
an(1/2*c)^2 - 4*tan(1/2*a)^2*tan(1/2*c)^2 - 25*tan(1/2*b*x + 1/2*c)*tan(1/2*a) + 4*tan(1/2*a)^2 + 25*tan(1/2*b
*x + 1/2*c)*tan(1/2*c) - 16*tan(1/2*a)*tan(1/2*c) + 4*tan(1/2*c)^2 - 4)/((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)^5))/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)^6,x)

[Out]

\text{Hanged}

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