3.1310 \(\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=212 \[ \frac{\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac{a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac{a \sin ^6(c+d x)}{6 b^2 d}+\frac{\sin ^7(c+d x)}{7 b d} \]

[Out]

-((a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^8*d)) + (a^2*(a^2 - b^2)^2*Sin[c + d*x])/(b^7*d) - (a*(a^2 -
b^2)^2*Sin[c + d*x]^2)/(2*b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^4)
/(4*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*b^3*d) - (a*Sin[c + d*x]^6)/(6*b^2*d) + Sin[c + d*x]^7/(7*b*d)

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Rubi [A]  time = 0.237986, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 948} \[ \frac{\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac{a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac{a \sin ^6(c+d x)}{6 b^2 d}+\frac{\sin ^7(c+d x)}{7 b d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]

[Out]

-((a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^8*d)) + (a^2*(a^2 - b^2)^2*Sin[c + d*x])/(b^7*d) - (a*(a^2 -
b^2)^2*Sin[c + d*x]^2)/(2*b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^4)
/(4*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*b^3*d) - (a*Sin[c + d*x]^6)/(6*b^2*d) + Sin[c + d*x]^7/(7*b*d)

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{\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (b^2-x^2\right )^2}{b^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a^3-a b^2\right )^2-a \left (a^2-b^2\right )^2 x+\left (a^2-b^2\right )^2 x^2-a \left (a^2-2 b^2\right ) x^3+\left (a^2-2 b^2\right ) x^4-a x^5+x^6-\frac{a^3 \left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=-\frac{a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}+\frac{a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac{a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{a \sin ^6(c+d x)}{6 b^2 d}+\frac{\sin ^7(c+d x)}{7 b d}\\ \end{align*}

Mathematica [A]  time = 1.24804, size = 180, normalized size = 0.85 \[ \frac{84 b^5 \left (a^2-2 b^2\right ) \sin ^5(c+d x)-105 a b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+140 b^3 \left (a^2-b^2\right )^2 \sin ^3(c+d x)-210 a b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+420 b \left (a^3-a b^2\right )^2 \sin (c+d x)-420 a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-70 a b^6 \sin ^6(c+d x)+60 b^7 \sin ^7(c+d x)}{420 b^8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]

[Out]

(-420*a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] + 420*b*(a^3 - a*b^2)^2*Sin[c + d*x] - 210*a*b^2*(a^2 - b^2)^2
*Sin[c + d*x]^2 + 140*b^3*(a^2 - b^2)^2*Sin[c + d*x]^3 - 105*a*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 84*b^5*(a^2
- 2*b^2)*Sin[c + d*x]^5 - 70*a*b^6*Sin[c + d*x]^6 + 60*b^7*Sin[c + d*x]^7)/(420*b^8*d)

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Maple [A]  time = 0.052, size = 329, normalized size = 1.6 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,bd}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,{b}^{2}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{5\,d{b}^{3}}}-{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,bd}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{3}}{4\,d{b}^{4}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\,{b}^{2}d}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d{b}^{5}}}-{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d{b}^{3}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}-{\frac{{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d{b}^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{d{b}^{4}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}+{\frac{{a}^{6}\sin \left ( dx+c \right ) }{d{b}^{7}}}-2\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d{b}^{5}}}+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d{b}^{3}}}-{\frac{{a}^{7}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{8}}}+2\,{\frac{{a}^{5}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{6}}}-{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x)

[Out]

1/7*sin(d*x+c)^7/b/d-1/6*a*sin(d*x+c)^6/b^2/d+1/5/d/b^3*sin(d*x+c)^5*a^2-2/5*sin(d*x+c)^5/b/d-1/4/d/b^4*sin(d*
x+c)^4*a^3+1/2*a*sin(d*x+c)^4/b^2/d+1/3/d/b^5*a^4*sin(d*x+c)^3-2/3/d/b^3*sin(d*x+c)^3*a^2+1/3*sin(d*x+c)^3/b/d
-1/2/d/b^6*sin(d*x+c)^2*a^5+1/d/b^4*sin(d*x+c)^2*a^3-1/2*a*sin(d*x+c)^2/b^2/d+1/d/b^7*a^6*sin(d*x+c)-2/d/b^5*s
in(d*x+c)*a^4+a^2*sin(d*x+c)/b^3/d-1/d*a^7/b^8*ln(a+b*sin(d*x+c))+2/d*a^5/b^6*ln(a+b*sin(d*x+c))-a^3*ln(a+b*si
n(d*x+c))/b^4/d

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Maxima [A]  time = 0.988877, size = 277, normalized size = 1.31 \begin{align*} \frac{\frac{60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \,{\left (a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \,{\left (a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (d x + c\right )^{4} + 140 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \,{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{2} + 420 \,{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac{420 \,{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/420*((60*b^6*sin(d*x + c)^7 - 70*a*b^5*sin(d*x + c)^6 + 84*(a^2*b^4 - 2*b^6)*sin(d*x + c)^5 - 105*(a^3*b^3 -
2*a*b^5)*sin(d*x + c)^4 + 140*(a^4*b^2 - 2*a^2*b^4 + b^6)*sin(d*x + c)^3 - 210*(a^5*b - 2*a^3*b^3 + a*b^5)*si
n(d*x + c)^2 + 420*(a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c))/b^7 - 420*(a^7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d
*x + c) + a)/b^8)/d

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Fricas [A]  time = 1.72782, size = 466, normalized size = 2.2 \begin{align*} \frac{70 \, a b^{6} \cos \left (d x + c\right )^{6} - 105 \, a^{3} b^{4} \cos \left (d x + c\right )^{4} + 210 \,{\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - 420 \,{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \,{\left (15 \, b^{7} \cos \left (d x + c\right )^{6} - 105 \, a^{6} b + 175 \, a^{4} b^{3} - 56 \, a^{2} b^{5} - 8 \, b^{7} - 3 \,{\left (7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} +{\left (35 \, a^{4} b^{3} - 28 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(70*a*b^6*cos(d*x + c)^6 - 105*a^3*b^4*cos(d*x + c)^4 + 210*(a^5*b^2 - a^3*b^4)*cos(d*x + c)^2 - 420*(a^
7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d*x + c) + a) - 4*(15*b^7*cos(d*x + c)^6 - 105*a^6*b + 175*a^4*b^3 - 56*a^2
*b^5 - 8*b^7 - 3*(7*a^2*b^5 + b^7)*cos(d*x + c)^4 + (35*a^4*b^3 - 28*a^2*b^5 - 4*b^7)*cos(d*x + c)^2)*sin(d*x
+ c))/(b^8*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(cos(d*x+c)**5*sin(d*x+c)**3/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.22209, size = 352, normalized size = 1.66 \begin{align*} \frac{\frac{60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, a^{2} b^{4} \sin \left (d x + c\right )^{5} - 168 \, b^{6} \sin \left (d x + c\right )^{5} - 105 \, a^{3} b^{3} \sin \left (d x + c\right )^{4} + 210 \, a b^{5} \sin \left (d x + c\right )^{4} + 140 \, a^{4} b^{2} \sin \left (d x + c\right )^{3} - 280 \, a^{2} b^{4} \sin \left (d x + c\right )^{3} + 140 \, b^{6} \sin \left (d x + c\right )^{3} - 210 \, a^{5} b \sin \left (d x + c\right )^{2} + 420 \, a^{3} b^{3} \sin \left (d x + c\right )^{2} - 210 \, a b^{5} \sin \left (d x + c\right )^{2} + 420 \, a^{6} \sin \left (d x + c\right ) - 840 \, a^{4} b^{2} \sin \left (d x + c\right ) + 420 \, a^{2} b^{4} \sin \left (d x + c\right )}{b^{7}} - \frac{420 \,{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/420*((60*b^6*sin(d*x + c)^7 - 70*a*b^5*sin(d*x + c)^6 + 84*a^2*b^4*sin(d*x + c)^5 - 168*b^6*sin(d*x + c)^5 -
105*a^3*b^3*sin(d*x + c)^4 + 210*a*b^5*sin(d*x + c)^4 + 140*a^4*b^2*sin(d*x + c)^3 - 280*a^2*b^4*sin(d*x + c)
^3 + 140*b^6*sin(d*x + c)^3 - 210*a^5*b*sin(d*x + c)^2 + 420*a^3*b^3*sin(d*x + c)^2 - 210*a*b^5*sin(d*x + c)^2
+ 420*a^6*sin(d*x + c) - 840*a^4*b^2*sin(d*x + c) + 420*a^2*b^4*sin(d*x + c))/b^7 - 420*(a^7 - 2*a^5*b^2 + a^
3*b^4)*log(abs(b*sin(d*x + c) + a))/b^8)/d