3.1313 \(\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=107 \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{b^3 d}-\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d} \]

[Out]

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

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Rubi [A]  time = 0.14031, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2837, 12, 894} \[ \frac{\left (a^2-2 b^2\right ) \sin (c+d x)}{b^3 d}-\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

Log[Sin[c + d*x]]/(a*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x])/(b^
3*d) - (a*Sin[c + d*x]^2)/(2*b^2*d) + Sin[c + d*x]^3/(3*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 894

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] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.139627, size = 101, normalized size = 0.94 \[ \frac{-3 a^2 b^2 \sin ^2(c+d x)+6 a b \left (a^2-2 b^2\right ) \sin (c+d x)+6 \left (b^4 \log (\sin (c+d x))-\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))\right )+2 a b^3 \sin ^3(c+d x)}{6 a b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.075, size = 140, normalized size = 1.3 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{{b}^{3}d}}-2\,{\frac{\sin \left ( dx+c \right ) }{bd}}-{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{4}d}}+2\,{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{da}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sin(d*x+c)^3/b/d-1/2*a*sin(d*x+c)^2/b^2/d+a^2*sin(d*x+c)/b^3/d-2*sin(d*x+c)/b/d-a^3*ln(a+b*sin(d*x+c))/b^4
/d+2/d/b^2*a*ln(a+b*sin(d*x+c))-1/d/a*ln(a+b*sin(d*x+c))+ln(sin(d*x+c))/a/d

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Maxima [A]  time = 0.989278, size = 134, normalized size = 1.25 \begin{align*} \frac{\frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a b^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*log(sin(d*x + c))/a + (2*b^2*sin(d*x + c)^3 - 3*a*b*sin(d*x + c)^2 + 6*(a^2 - 2*b^2)*sin(d*x + c))/b^3
- 6*(a^4 - 2*a^2*b^2 + b^4)*log(b*sin(d*x + c) + a)/(a*b^4))/d

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Fricas [A]  time = 1.75831, size = 250, normalized size = 2.34 \begin{align*} \frac{3 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 6 \, b^{4} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{6 \, a b^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.19394, size = 143, normalized size = 1.34 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right ) - 12 \, b^{2} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a b^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*log(abs(sin(d*x + c)))/a + (2*b^2*sin(d*x + c)^3 - 3*a*b*sin(d*x + c)^2 + 6*a^2*sin(d*x + c) - 12*b^2*s
in(d*x + c))/b^3 - 6*(a^4 - 2*a^2*b^2 + b^4)*log(abs(b*sin(d*x + c) + a))/(a*b^4))/d