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

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

[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)

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Rubi [A]  time = 0.299352, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2837, 12, 948} \[ \frac{\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}-\frac{2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}-\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \cos (c+d x)}{a^6 d}+\frac{b^2 \left (a^2-b^2\right )^2}{a^7 d (a \cos (c+d x)+b)}+\frac{2 b \left (-4 a^2 b^2+a^4+3 b^4\right ) \log (a \cos (c+d x)+b)}{a^7 d}+\frac{b \cos ^4(c+d x)}{2 a^3 d}-\frac{\cos ^5(c+d x)}{5 a^2 d} \]

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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2837

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*x)/b)^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 12

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

Rule 948

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]))

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{\operatorname{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{\operatorname{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{\operatorname{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*}

Mathematica [A]  time = 1.07203, size = 280, normalized size = 1.44 \[ \frac{-30 a^4 b^2 \cos (4 (c+d x))+120 a^3 b^3 \cos (3 (c+d x))-5 \left (-168 a^4 b^2+144 a^2 b^4+25 a^6\right ) \cos (2 (c+d x))+960 a^4 b^2 \log (a \cos (c+d x)+b)-3840 a^2 b^4 \log (a \cos (c+d x)+b)+120 a b \cos (c+d x) \left (8 \left (-4 a^2 b^2+a^4+3 b^4\right ) \log (a \cos (c+d x)+b)+23 a^2 b^2-4 a^4-20 b^4\right )+1740 a^4 b^2-2160 a^2 b^4-115 a^5 b \cos (3 (c+d x))+9 a^5 b \cos (5 (c+d x))+22 a^6 \cos (4 (c+d x))-3 a^6 \cos (6 (c+d x))-150 a^6+2880 b^6 \log (a \cos (c+d x)+b)+480 b^6}{480 a^7 d (a \cos (c+d x)+b)} \]

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 = 0.061, size = 285, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,{a}^{2}d}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2\,{a}^{3}d}}+{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{2}d}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{d{a}^{4}}}-2\,{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{{a}^{3}d}}+2\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{d{a}^{5}}}-{\frac{\cos \left ( dx+c \right ) }{{a}^{2}d}}+6\,{\frac{{b}^{2}\cos \left ( dx+c \right ) }{d{a}^{4}}}-5\,{\frac{{b}^{4}\cos \left ( dx+c \right ) }{d{a}^{6}}}+2\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{{a}^{3}d}}-8\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{5}}}+6\,{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{7}}}+{\frac{{b}^{2}}{{a}^{3}d \left ( b+a\cos \left ( dx+c \right ) \right ) }}-2\,{\frac{{b}^{4}}{d{a}^{5} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+{\frac{{b}^{6}}{d{a}^{7} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.04436, size = 248, normalized size = 1.28 \begin{align*} \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{align*}

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 = 2.13182, size = 563, normalized size = 2.9 \begin{align*} -\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{align*}

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]  time = 1.37862, size = 1488, normalized size = 7.67 \begin{align*} \text{result too large to display} \end{align*}

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