3.3.3 \(\int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx\) [203]

Optimal. Leaf size=230 \[ \frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) x}{16 a^7}-\frac {2 (a-b)^{5/2} b (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d}+\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d} \]

[Out]

1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*x/a^7-2*(a-b)^(5/2)*b*(a+b)^(5/2)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/
2*c)/(a+b)^(1/2))/a^7/d+1/16*(16*b*(a^2-b^2)^2-a*(5*a^4-14*a^2*b^2+8*b^4)*cos(d*x+c))*sin(d*x+c)/a^6/d+1/24*(8
*b*(a^2-b^2)-a*(5*a^2-6*b^2)*cos(d*x+c))*sin(d*x+c)^3/a^4/d+1/30*(6*b-5*a*cos(d*x+c))*sin(d*x+c)^5/a^2/d

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Rubi [A]
time = 0.41, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2944, 2814, 2738, 214} \begin {gather*} -\frac {2 b (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d}+\frac {\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}+\frac {\sin ^3(c+d x) \left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right )}{24 a^4 d}+\frac {\sin (c+d x) \left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right )}{16 a^6 d}+\frac {x \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )}{16 a^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x]),x]

[Out]

((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*x)/(16*a^7) - (2*(a - b)^(5/2)*b*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b
]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^7*d) + ((16*b*(a^2 - b^2)^2 - a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*Cos[c + d*x]
)*Sin[c + d*x])/(16*a^6*d) + ((8*b*(a^2 - b^2) - a*(5*a^2 - 6*b^2)*Cos[c + d*x])*Sin[c + d*x]^3)/(24*a^4*d) +
((6*b - 5*a*Cos[c + d*x])*Sin[c + d*x]^5)/(30*a^2*d)

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2944

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + p)*(m + p +
1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^6(c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}-\frac {\int \frac {\left (-a b+\left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^4(c+d x)}{-b-a \cos (c+d x)} \, dx}{6 a^2}\\ &=\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}-\frac {\int \frac {\left (-3 a b \left (3 a^2-2 b^2\right )+3 \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{24 a^4}\\ &=\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}-\frac {\int \frac {-3 a b \left (11 a^4-18 a^2 b^2+8 b^4\right )+3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{48 a^6}\\ &=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) x}{16 a^7}+\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}+\frac {\left (b \left (a^2-b^2\right )^3\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a^7}\\ &=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) x}{16 a^7}+\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}+\frac {\left (2 b \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) x}{16 a^7}-\frac {2 (a-b)^{5/2} b (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d}+\frac {\left (16 b \left (a^2-b^2\right )^2-a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sin (c+d x)}{16 a^6 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (5 a^2-6 b^2\right ) \cos (c+d x)\right ) \sin ^3(c+d x)}{24 a^4 d}+\frac {(6 b-5 a \cos (c+d x)) \sin ^5(c+d x)}{30 a^2 d}\\ \end {align*}

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Mathematica [A]
time = 1.57, size = 268, normalized size = 1.17 \begin {gather*} \frac {300 a^6 c-1800 a^4 b^2 c+2400 a^2 b^4 c-960 b^6 c+300 a^6 d x-1800 a^4 b^2 d x+2400 a^2 b^4 d x-960 b^6 d x+1920 b \left (a^2-b^2\right )^{5/2} \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+120 a b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin (c+d x)-15 \left (15 a^6-32 a^4 b^2+16 a^2 b^4\right ) \sin (2 (c+d x))-140 a^5 b \sin (3 (c+d x))+80 a^3 b^3 \sin (3 (c+d x))+45 a^6 \sin (4 (c+d x))-30 a^4 b^2 \sin (4 (c+d x))+12 a^5 b \sin (5 (c+d x))-5 a^6 \sin (6 (c+d x))}{960 a^7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x]),x]

[Out]

(300*a^6*c - 1800*a^4*b^2*c + 2400*a^2*b^4*c - 960*b^6*c + 300*a^6*d*x - 1800*a^4*b^2*d*x + 2400*a^2*b^4*d*x -
960*b^6*d*x + 1920*b*(a^2 - b^2)^(5/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 120*a*b*(11*a^4
- 18*a^2*b^2 + 8*b^4)*Sin[c + d*x] - 15*(15*a^6 - 32*a^4*b^2 + 16*a^2*b^4)*Sin[2*(c + d*x)] - 140*a^5*b*Sin[3
*(c + d*x)] + 80*a^3*b^3*Sin[3*(c + d*x)] + 45*a^6*Sin[4*(c + d*x)] - 30*a^4*b^2*Sin[4*(c + d*x)] + 12*a^5*b*S
in[5*(c + d*x)] - 5*a^6*Sin[6*(c + d*x)])/(960*a^7*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(213)=426\).
time = 0.19, size = 441, normalized size = 1.92 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2*(a-b)^3*(a+b)^3*b/a^7/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+2/a^7*
(((5/16*a^6+a^5*b-7/8*a^4*b^2-2*a^3*b^3+1/2*a^2*b^4+a*b^5)*tan(1/2*d*x+1/2*c)^11+(19/3*a^5*b-29/8*a^4*b^2+3/2*
a^2*b^4+5*a*b^5+85/48*a^6-34/3*a^3*b^3)*tan(1/2*d*x+1/2*c)^9+(86/5*a^5*b-11/4*a^4*b^2-24*a^3*b^3+a^2*b^4+10*a*
b^5+33/8*a^6)*tan(1/2*d*x+1/2*c)^7+(-33/8*a^6+11/4*a^4*b^2-a^2*b^4+86/5*a^5*b-24*a^3*b^3+10*a*b^5)*tan(1/2*d*x
+1/2*c)^5+(19/3*a^5*b+29/8*a^4*b^2-34/3*a^3*b^3-3/2*a^2*b^4+5*a*b^5-85/48*a^6)*tan(1/2*d*x+1/2*c)^3+(a^5*b-2*a
^3*b^3+a*b^5-5/16*a^6+7/8*a^4*b^2-1/2*a^2*b^4)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(5*a^6-30*a
^4*b^2+40*a^2*b^4-16*b^6)*arctan(tan(1/2*d*x+1/2*c))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [A]
time = 3.15, size = 553, normalized size = 2.40 \begin {gather*} \left [\frac {15 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} d x + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (40 \, a^{6} \cos \left (d x + c\right )^{5} - 48 \, a^{5} b \cos \left (d x + c\right )^{4} - 368 \, a^{5} b + 560 \, a^{3} b^{3} - 240 \, a b^{5} - 10 \, {\left (13 \, a^{6} - 6 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (11 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, a^{6} - 18 \, a^{4} b^{2} + 8 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{7} d}, \frac {15 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} d x - 240 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (40 \, a^{6} \cos \left (d x + c\right )^{5} - 48 \, a^{5} b \cos \left (d x + c\right )^{4} - 368 \, a^{5} b + 560 \, a^{3} b^{3} - 240 \, a b^{5} - 10 \, {\left (13 \, a^{6} - 6 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (11 \, a^{5} b - 5 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, a^{6} - 18 \, a^{4} b^{2} + 8 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{7} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/240*(15*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*d*x + 120*(a^4*b - 2*a^2*b^3 + b^5)*sqrt(a^2 - b^2)*log(
(2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a
^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - (40*a^6*cos(d*x + c)^5 - 48*a^5*b*cos(d*x + c)^4
- 368*a^5*b + 560*a^3*b^3 - 240*a*b^5 - 10*(13*a^6 - 6*a^4*b^2)*cos(d*x + c)^3 + 16*(11*a^5*b - 5*a^3*b^3)*cos
(d*x + c)^2 + 15*(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(a^7*d), 1/240*(15*(5*a^6 - 30*
a^4*b^2 + 40*a^2*b^4 - 16*b^6)*d*x - 240*(a^4*b - 2*a^2*b^3 + b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(
b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (40*a^6*cos(d*x + c)^5 - 48*a^5*b*cos(d*x + c)^4 - 368*a^5*b
+ 560*a^3*b^3 - 240*a*b^5 - 10*(13*a^6 - 6*a^4*b^2)*cos(d*x + c)^3 + 16*(11*a^5*b - 5*a^3*b^3)*cos(d*x + c)^2
+ 15*(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(a^7*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{6}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**6/(a+b*sec(d*x+c)),x)

[Out]

Integral(sin(c + d*x)**6/(a + b*sec(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (212) = 424\).
time = 0.49, size = 781, normalized size = 3.40 \begin {gather*} \frac {\frac {15 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} {\left (d x + c\right )}}{a^{7}} - \frac {480 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{7}} + \frac {2 \, {\left (75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 210 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 480 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 120 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1520 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 870 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2720 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4128 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 660 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5760 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 240 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4128 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5760 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1520 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 870 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2720 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 360 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 210 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 480 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{6}}}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(15*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*(d*x + c)/a^7 - 480*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)
*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))
/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^7) + 2*(75*a^5*tan(1/2*d*x + 1/2*c)^11 + 240*a^4*b*tan(1/2*d*x + 1/2*c
)^11 - 210*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 - 480*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*a*b^4*tan(1/2*d*x + 1/2
*c)^11 + 240*b^5*tan(1/2*d*x + 1/2*c)^11 + 425*a^5*tan(1/2*d*x + 1/2*c)^9 + 1520*a^4*b*tan(1/2*d*x + 1/2*c)^9
- 870*a^3*b^2*tan(1/2*d*x + 1/2*c)^9 - 2720*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 + 360*a*b^4*tan(1/2*d*x + 1/2*c)^9
+ 1200*b^5*tan(1/2*d*x + 1/2*c)^9 + 990*a^5*tan(1/2*d*x + 1/2*c)^7 + 4128*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 660*a
^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 5760*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 240*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 2400*
b^5*tan(1/2*d*x + 1/2*c)^7 - 990*a^5*tan(1/2*d*x + 1/2*c)^5 + 4128*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 660*a^3*b^2*
tan(1/2*d*x + 1/2*c)^5 - 5760*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 240*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 2400*b^5*tan
(1/2*d*x + 1/2*c)^5 - 425*a^5*tan(1/2*d*x + 1/2*c)^3 + 1520*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 870*a^3*b^2*tan(1/2
*d*x + 1/2*c)^3 - 2720*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 360*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 1200*b^5*tan(1/2*d*
x + 1/2*c)^3 - 75*a^5*tan(1/2*d*x + 1/2*c) + 240*a^4*b*tan(1/2*d*x + 1/2*c) + 210*a^3*b^2*tan(1/2*d*x + 1/2*c)
- 480*a^2*b^3*tan(1/2*d*x + 1/2*c) - 120*a*b^4*tan(1/2*d*x + 1/2*c) + 240*b^5*tan(1/2*d*x + 1/2*c))/((tan(1/2
*d*x + 1/2*c)^2 + 1)^6*a^6))/d

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Mupad [B]
time = 3.88, size = 2500, normalized size = 10.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^6/(a + b/cos(c + d*x)),x)

[Out]

(atan(((((((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4 - 134*a
^19*b^3 + 6*a^20*b^2)/a^18 - (tan(c/2 + (d*x)/2)*(512*a^16*b + 512*a^14*b^3 - 1024*a^15*b^2)*(a^6*5i - b^6*16i
+ a^2*b^4*40i - a^4*b^2*30i))/(128*a^19))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7) + (tan(c/2
+ (d*x)/2)*(1024*a*b^14 - 75*a^14*b + 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10
240*a^5*b^10 - 10240*a^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31
*a^13*b^2))/(8*a^12))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i)*1i)/(16*a^7) - (((((42*a^21*b - 10*a^22 +
32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4 - 134*a^19*b^3 + 6*a^20*b^2)/a^18 + (tan
(c/2 + (d*x)/2)*(512*a^16*b + 512*a^14*b^3 - 1024*a^15*b^2)*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(1
28*a^19))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7) - (tan(c/2 + (d*x)/2)*(1024*a*b^14 - 75*a^1
4*b + 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10240*a^5*b^10 - 10240*a^7*b^8 + 25
40*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b^2))/(8*a^12))*(a^6*5i - b
^6*16i + a^2*b^4*40i - a^4*b^2*30i)*1i)/(16*a^7))/(((25*a^19*b)/4 - 96*a*b^19 + 64*b^20 - 480*a^2*b^18 + 760*a
^3*b^17 + 1544*a^4*b^16 - 2628*a^5*b^15 - 2748*a^6*b^14 + 5179*a^7*b^13 + 2890*a^8*b^12 - 6359*a^9*b^11 - 1736
*a^10*b^10 + (19951*a^11*b^9)/4 + (937*a^12*b^8)/2 - (4915*a^13*b^7)/2 + (85*a^14*b^6)/2 + 715*a^15*b^5 - (105
*a^16*b^4)/2 - (215*a^17*b^3)/2 + (15*a^18*b^2)/2)/a^18 + (((((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7
- 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4 - 134*a^19*b^3 + 6*a^20*b^2)/a^18 - (tan(c/2 + (d*x)/2)*(512*a^16*
b + 512*a^14*b^3 - 1024*a^15*b^2)*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(128*a^19))*(a^6*5i - b^6*16
i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7) + (tan(c/2 + (d*x)/2)*(1024*a*b^14 - 75*a^14*b + 25*a^15 - 512*b^15 +
2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10240*a^5*b^10 - 10240*a^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6
- 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b^2))/(8*a^12))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4
*b^2*30i))/(16*a^7) + (((((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a
^18*b^4 - 134*a^19*b^3 + 6*a^20*b^2)/a^18 + (tan(c/2 + (d*x)/2)*(512*a^16*b + 512*a^14*b^3 - 1024*a^15*b^2)*(a
^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(128*a^19))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*
a^7) - (tan(c/2 + (d*x)/2)*(1024*a*b^14 - 75*a^14*b + 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 256
0*a^4*b^11 + 10240*a^5*b^10 - 10240*a^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 61
9*a^12*b^3 + 31*a^13*b^2))/(8*a^12))*(a^6*5i - b^6*16i + a^2*b^4*40i - a^4*b^2*30i))/(16*a^7)))*(a^6*5i - b^6*
16i + a^2*b^4*40i - a^4*b^2*30i)*1i)/(8*a^7*d) - ((tan(c/2 + (d*x)/2)*(8*a*b^4 - 16*a^4*b + 5*a^5 - 16*b^5 + 3
2*a^2*b^3 - 14*a^3*b^2))/(8*a^6) - (tan(c/2 + (d*x)/2)^11*(8*a*b^4 + 16*a^4*b + 5*a^5 + 16*b^5 - 32*a^2*b^3 -
14*a^3*b^2))/(8*a^6) + (tan(c/2 + (d*x)/2)^3*(72*a*b^4 - 304*a^4*b + 85*a^5 - 240*b^5 + 544*a^2*b^3 - 174*a^3*
b^2))/(24*a^6) - (tan(c/2 + (d*x)/2)^9*(72*a*b^4 + 304*a^4*b + 85*a^5 + 240*b^5 - 544*a^2*b^3 - 174*a^3*b^2))/
(24*a^6) + (tan(c/2 + (d*x)/2)^5*(40*a*b^4 - 688*a^4*b + 165*a^5 - 400*b^5 + 960*a^2*b^3 - 110*a^3*b^2))/(20*a
^6) - (tan(c/2 + (d*x)/2)^7*(40*a*b^4 + 688*a^4*b + 165*a^5 + 400*b^5 - 960*a^2*b^3 - 110*a^3*b^2))/(20*a^6))/
(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*t
an(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (b*atan(((b*((tan(c/2 + (d*x)/2)*(1024*a*b^14 - 75*a^14*b
+ 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10240*a^5*b^10 - 10240*a^7*b^8 + 2540*
a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b^2))/(8*a^12) + (b*((a + b)^5
*(a - b)^5)^(1/2)*((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5 + 52*a^18*b^4
- 134*a^19*b^3 + 6*a^20*b^2)/a^18 - (b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(512*a^16*b + 512*a^14*
b^3 - 1024*a^15*b^2))/(8*a^19)))/a^7)*((a + b)^5*(a - b)^5)^(1/2)*1i)/a^7 + (b*((tan(c/2 + (d*x)/2)*(1024*a*b^
14 - 75*a^14*b + 25*a^15 - 512*b^15 + 2048*a^2*b^13 - 5120*a^3*b^12 - 2560*a^4*b^11 + 10240*a^5*b^10 - 10240*a
^7*b^8 + 2540*a^8*b^7 + 5180*a^9*b^6 - 2064*a^10*b^5 - 1136*a^11*b^4 + 619*a^12*b^3 + 31*a^13*b^2))/(8*a^12) -
(b*((a + b)^5*(a - b)^5)^(1/2)*((42*a^21*b - 10*a^22 + 32*a^14*b^8 - 48*a^15*b^7 - 80*a^16*b^6 + 140*a^17*b^5
+ 52*a^18*b^4 - 134*a^19*b^3 + 6*a^20*b^2)/a^18 + (b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(512*a^16
*b + 512*a^14*b^3 - 1024*a^15*b^2))/(8*a^19)))/a^7)*((a + b)^5*(a - b)^5)^(1/2)*1i)/a^7)/(((25*a^19*b)/4 - 96*
a*b^19 + 64*b^20 - 480*a^2*b^18 + 760*a^3*b^17 + 1544*a^4*b^16 - 2628*a^5*b^15 - 2748*a^6*b^14 + 5179*a^7*b^13
+ 2890*a^8*b^12 - 6359*a^9*b^11 - 1736*a^10*b^...

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