3.3.10 \(\int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) [210]

Optimal. Leaf size=194 \[ -\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (b+a \cos (c+d x))}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{a^7 d} \]

[Out]

-(a^4-6*a^2*b^2+5*b^4)*cos(d*x+c)/a^6/d-2*b*(a^2-b^2)*cos(d*x+c)^2/a^5/d+1/3*(2*a^2-3*b^2)*cos(d*x+c)^3/a^4/d+
1/2*b*cos(d*x+c)^4/a^3/d-1/5*cos(d*x+c)^5/a^2/d+b^2*(a^2-b^2)^2/a^7/d/(b+a*cos(d*x+c))+2*b*(a^4-4*a^2*b^2+3*b^
4)*ln(b+a*cos(d*x+c))/a^7/d

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Rubi [A]
time = 0.21, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 962} \begin {gather*} \frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (a \cos (c+d x)+b)}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b)}{a^7 d}-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

-(((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x])/(a^6*d)) - (2*b*(a^2 - b^2)*Cos[c + d*x]^2)/(a^5*d) + ((2*a^2 - 3*b
^2)*Cos[c + d*x]^3)/(3*a^4*d) + (b*Cos[c + d*x]^4)/(2*a^3*d) - Cos[c + d*x]^5/(5*a^2*d) + (b^2*(a^2 - b^2)^2)/
(a^7*d*(b + a*Cos[c + d*x])) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]])/(a^7*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 962

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rule 2916

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

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 ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^5(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (a^4 \left (1+\frac {-6 a^2 b^2+5 b^4}{a^4}\right )+\frac {b^2 \left (a^2-b^2\right )^2}{(b-x)^2}-\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right )}{b-x}+4 b \left (-a^2+b^2\right ) x-\left (2 a^2-3 b^2\right ) x^2+2 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (b+a \cos (c+d x))}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{a^7 d}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 280, normalized size = 1.44 \begin {gather*} \frac {-150 a^6+1740 a^4 b^2-2160 a^2 b^4+480 b^6-5 \left (25 a^6-168 a^4 b^2+144 a^2 b^4\right ) \cos (2 (c+d x))-115 a^5 b \cos (3 (c+d x))+120 a^3 b^3 \cos (3 (c+d x))+22 a^6 \cos (4 (c+d x))-30 a^4 b^2 \cos (4 (c+d x))+9 a^5 b \cos (5 (c+d x))-3 a^6 \cos (6 (c+d x))+960 a^4 b^2 \log (b+a \cos (c+d x))-3840 a^2 b^4 \log (b+a \cos (c+d x))+2880 b^6 \log (b+a \cos (c+d x))+120 a b \cos (c+d x) \left (-4 a^4+23 a^2 b^2-20 b^4+8 \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))\right )}{480 a^7 d (b+a \cos (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

(-150*a^6 + 1740*a^4*b^2 - 2160*a^2*b^4 + 480*b^6 - 5*(25*a^6 - 168*a^4*b^2 + 144*a^2*b^4)*Cos[2*(c + d*x)] -
115*a^5*b*Cos[3*(c + d*x)] + 120*a^3*b^3*Cos[3*(c + d*x)] + 22*a^6*Cos[4*(c + d*x)] - 30*a^4*b^2*Cos[4*(c + d*
x)] + 9*a^5*b*Cos[5*(c + d*x)] - 3*a^6*Cos[6*(c + d*x)] + 960*a^4*b^2*Log[b + a*Cos[c + d*x]] - 3840*a^2*b^4*L
og[b + a*Cos[c + d*x]] + 2880*b^6*Log[b + a*Cos[c + d*x]] + 120*a*b*Cos[c + d*x]*(-4*a^4 + 23*a^2*b^2 - 20*b^4
+ 8*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]]))/(480*a^7*d*(b + a*Cos[c + d*x]))

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Maple [A]
time = 0.20, size = 198, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{2}-\frac {2 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{2} b^{2} \left (\cos ^{3}\left (d x +c \right )\right )+2 a^{3} b \left (\cos ^{2}\left (d x +c \right )\right )-2 a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right )+a^{4} \cos \left (d x +c \right )-6 a^{2} b^{2} \cos \left (d x +c \right )+5 b^{4} \cos \left (d x +c \right )}{a^{6}}+\frac {b^{2} \left (a^{4}-2 b^{2} a^{2}+b^{4}\right )}{a^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}-4 b^{2} a^{2}+3 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{7}}}{d}\)	\(198\)
default	\(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{2}-\frac {2 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{2} b^{2} \left (\cos ^{3}\left (d x +c \right )\right )+2 a^{3} b \left (\cos ^{2}\left (d x +c \right )\right )-2 a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right )+a^{4} \cos \left (d x +c \right )-6 a^{2} b^{2} \cos \left (d x +c \right )+5 b^{4} \cos \left (d x +c \right )}{a^{6}}+\frac {b^{2} \left (a^{4}-2 b^{2} a^{2}+b^{4}\right )}{a^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}-4 b^{2} a^{2}+3 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{7}}}{d}\)	\(198\)
risch	\(\frac {2 b^{2} \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{7} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {8 i b^{3} x}{a^{5}}-\frac {6 i b^{5} x}{a^{7}}-\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{5} d}+\frac {21 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{8 a^{4} d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 a^{6} d}+\frac {16 i b^{3} c}{a^{5} d}-\frac {12 i b^{5} c}{a^{7} d}+\frac {21 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{8 a^{4} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 a^{6} d}-\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{5} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}-\frac {8 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{5} d}+\frac {6 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{7} d}+\frac {5 \cos \left (3 d x +3 c \right )}{48 d \,a^{2}}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{2} d}+\frac {b \cos \left (4 d x +4 c \right )}{16 d \,a^{3}}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{4 d \,a^{4}}-\frac {2 i b x}{a^{3}}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{2}}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{2} d}-\frac {4 i b c}{a^{3} d}\)	\(503\)
norman	\(\frac {\frac {\left (32 a^{5}+32 a^{4} b +96 a^{3} b^{2}-360 a^{2} b^{3}-144 a \,b^{4}+360 b^{5}\right ) \left (a +b \right )}{60 a^{6} b d}-\frac {\left (32 a^{6}-128 a^{5} b +120 a^{3} b^{3}-408 a^{2} b^{4}-72 a \,b^{5}+360 b^{6}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{6} b d}+\frac {\left (32 a^{6}+128 a^{5} b -120 a^{3} b^{3}-408 a^{2} b^{4}+72 a \,b^{5}+360 b^{6}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{6} b d}-\frac {\left (32 a^{6}-64 a^{5} b +128 a^{4} b^{2}+264 a^{3} b^{3}-504 a^{2} b^{4}-216 a \,b^{5}+360 b^{6}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a^{6} d b}-\frac {\left (64 a^{6}-160 a^{5} b +168 a^{4} b^{2}+432 a^{3} b^{3}-888 a^{2} b^{4}-288 a \,b^{5}+720 b^{6}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 a^{6} b d}+\frac {\left (64 a^{6}+160 a^{5} b +168 a^{4} b^{2}-432 a^{3} b^{3}-888 a^{2} b^{4}+288 a \,b^{5}+720 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 a^{6} d b}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {2 b \left (a^{4}-4 b^{2} a^{2}+3 b^{4}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{7} d}+\frac {2 b \left (a^{4}-4 b^{2} a^{2}+3 b^{4}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{a^{7} d}\)	\(538\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^5/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/a^6*(1/5*cos(d*x+c)^5*a^4-1/2*b*cos(d*x+c)^4*a^3-2/3*a^4*cos(d*x+c)^3+a^2*b^2*cos(d*x+c)^3+2*a^3*b*cos
(d*x+c)^2-2*a*b^3*cos(d*x+c)^2+a^4*cos(d*x+c)-6*a^2*b^2*cos(d*x+c)+5*b^4*cos(d*x+c))+b^2*(a^4-2*a^2*b^2+b^4)/a
^7/(b+a*cos(d*x+c))+2*b/a^7*(a^4-4*a^2*b^2+3*b^4)*ln(b+a*cos(d*x+c)))

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Maxima [A]
time = 0.28, size = 184, normalized size = 0.95 \begin {gather*} \frac {\frac {30 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )}}{a^{8} \cos \left (d x + c\right ) + a^{7} b} - \frac {6 \, a^{4} \cos \left (d x + c\right )^{5} - 15 \, a^{3} b \cos \left (d x + c\right )^{4} - 10 \, {\left (2 \, a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )}{a^{6}} + \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{7}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

1/30*(30*(a^4*b^2 - 2*a^2*b^4 + b^6)/(a^8*cos(d*x + c) + a^7*b) - (6*a^4*cos(d*x + c)^5 - 15*a^3*b*cos(d*x + c
)^4 - 10*(2*a^4 - 3*a^2*b^2)*cos(d*x + c)^3 + 60*(a^3*b - a*b^3)*cos(d*x + c)^2 + 30*(a^4 - 6*a^2*b^2 + 5*b^4)
*cos(d*x + c))/a^6 + 60*(a^4*b - 4*a^2*b^3 + 3*b^5)*log(a*cos(d*x + c) + b)/a^7)/d

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Fricas [A]
time = 2.61, size = 240, normalized size = 1.24 \begin {gather*} -\frac {48 \, a^{6} \cos \left (d x + c\right )^{6} - 72 \, a^{5} b \cos \left (d x + c\right )^{5} - 435 \, a^{4} b^{2} + 720 \, a^{2} b^{4} - 240 \, b^{6} - 40 \, {\left (4 \, a^{6} - 3 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3} + 240 \, {\left (a^{6} - 4 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, a^{5} b - 80 \, a^{3} b^{3} + 80 \, a b^{5}\right )} \cos \left (d x + c\right ) - 480 \, {\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6} + {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{240 \, {\left (a^{8} d \cos \left (d x + c\right ) + a^{7} b d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/240*(48*a^6*cos(d*x + c)^6 - 72*a^5*b*cos(d*x + c)^5 - 435*a^4*b^2 + 720*a^2*b^4 - 240*b^6 - 40*(4*a^6 - 3*
a^4*b^2)*cos(d*x + c)^4 + 80*(4*a^5*b - 3*a^3*b^3)*cos(d*x + c)^3 + 240*(a^6 - 4*a^4*b^2 + 3*a^2*b^4)*cos(d*x
+ c)^2 + 15*(3*a^5*b - 80*a^3*b^3 + 80*a*b^5)*cos(d*x + c) - 480*(a^4*b^2 - 4*a^2*b^4 + 3*b^6 + (a^5*b - 4*a^3
*b^3 + 3*a*b^5)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^8*d*cos(d*x + c) + a^7*b*d)

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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(sin(d*x+c)**5/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (188) = 376\).
time = 0.48, size = 1102, normalized size = 5.68 \begin {gather*} \frac {\frac {60 \, {\left (a^{5} b - a^{4} b^{2} - 4 \, a^{3} b^{3} + 4 \, a^{2} b^{4} + 3 \, a b^{5} - 3 \, b^{6}\right )} \log \left ({\left | a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - a^{7} b} - \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{7}} - \frac {60 \, {\left (a^{5} b - 5 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + 4 \, a b^{5} + 3 \, b^{6} + \frac {a^{5} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{4} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, a^{3} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{2} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} a^{7}} + \frac {32 \, a^{5} - 137 \, a^{4} b - 300 \, a^{3} b^{2} + 548 \, a^{2} b^{3} + 300 \, a b^{4} - 411 \, b^{5} - \frac {160 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {805 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1320 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2980 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {1200 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2055 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {320 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1970 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1920 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6200 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1800 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4110 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1970 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1080 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6200 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1200 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4110 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {805 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {180 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2980 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {300 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2055 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {137 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {548 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {411 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{7} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/30*(60*(a^5*b - a^4*b^2 - 4*a^3*b^3 + 4*a^2*b^4 + 3*a*b^5 - 3*b^6)*log(abs(a + b + a*(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^8 - a^7*b) - 60*(a^4*b - 4*a^2*b^3 + 3*b^5)*log(
abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^7 - 60*(a^5*b - 5*a^3*b^3 - 3*a^2*b^4 + 4*a*b^5 + 3*b^6 + a
^5*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a^4*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*a^3*b^3*(cos(d*
x + c) - 1)/(cos(d*x + c) + 1) + 4*a^2*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a*b^5*(cos(d*x + c) - 1)/
(cos(d*x + c) + 1) - 3*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*a^7) + (32*a^5 - 137*a^4*b - 300*a^3*b^2 + 548*a^2*b^3 + 300*
a*b^4 - 411*b^5 - 160*a^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 805*a^4*b*(cos(d*x + c) - 1)/(cos(d*x + c) +
1) + 1320*a^3*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2980*a^2*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
- 1200*a*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2055*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 320*a^5*
(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1970*a^4*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1920*a^3*b^
2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 6200*a^2*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1800*a*
b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 4110*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1970*a^4*
b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 1080*a^3*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 6200*a^
2*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 1200*a*b^4*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 4110*
b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 805*a^4*b*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 180*a^3*
b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 2980*a^2*b^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 300*a
*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 2055*b^5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 137*a^4*
b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 548*a^2*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 411*b^5*
(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/(a^7*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5))/d

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Mupad [B]
time = 0.12, size = 253, normalized size = 1.30 \begin {gather*} \frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,a^2}-\frac {b^2}{a^4}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^3}{a^5}+\frac {b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2\,a^3\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^4\,b-8\,a^2\,b^3+6\,b^5\right )}{a^7\,d}+\frac {a^4\,b^2-2\,a^2\,b^4+b^6}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^7+b\,a^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^5/(a + b/cos(c + d*x))^2,x)

[Out]

(cos(c + d*x)^3*(2/(3*a^2) - b^2/a^4))/d - (cos(c + d*x)^2*(b^3/a^5 + (b*(2/a^2 - (3*b^2)/a^4))/a))/d - (cos(c
+ d*x)*(1/a^2 + (b^2*(2/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^3)/a^5 + (2*b*(2/a^2 - (3*b^2)/a^4))/a))/a))/d -
cos(c + d*x)^5/(5*a^2*d) + (b*cos(c + d*x)^4)/(2*a^3*d) + (log(b + a*cos(c + d*x))*(2*a^4*b + 6*b^5 - 8*a^2*b
^3))/(a^7*d) + (b^6 - 2*a^2*b^4 + a^4*b^2)/(a*d*(a^7*cos(c + d*x) + a^6*b))

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