3.1316 \(\int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=120 \[ \frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d} \]

[Out]

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

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Rubi [A]  time = 0.174388, antiderivative size = 120, 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} \[ \frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^2 - b^2)*Csc[c + d*x])/(a^3*d) + (b*Csc[c + d*x]^2)/(2*a^2*d) - Csc[c + d*x]^3/(3*a*d) + (b*(2*a^2 - b^2
)*Log[Sin[c + d*x]])/(a^4*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^4*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 (c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^4 \left (b^2-x^2\right )^2}{x^4 (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^4 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{b^4}{a x^4}-\frac{b^4}{a^2 x^3}+\frac{-2 a^2 b^2+b^4}{a^3 x^2}+\frac{2 a^2 b^2-b^4}{a^4 x}+\frac{\left (a^2-b^2\right )^2}{a^4 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\left (2 a^2-b^2\right ) \csc (c+d x)}{a^3 d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{b \left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^4 b d}\\ \end{align*}

Mathematica [A]  time = 0.283179, size = 110, normalized size = 0.92 \[ \frac{3 a^2 b^2 \csc ^2(c+d x)+6 a b \left (2 a^2-b^2\right ) \csc (c+d x)-6 b^2 \left (b^2-2 a^2\right ) \log (\sin (c+d x))+6 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-2 a^3 b \csc ^3(c+d x)}{6 a^4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.093, size = 163, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}}-2\,{\frac{b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{{b}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+2\,{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\sin \left ( dx+c \right ) }}+2\,{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{b}{2\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^4/(a+b*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.974852, size = 153, normalized size = 1.27 \begin{align*} \frac{\frac{6 \,{\left (2 \, a^{2} b - b^{3}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}} + \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^{4} b} + \frac{3 \, a b \sin \left (d x + c\right ) + 6 \,{\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - 2 \, a^{2}}{a^{3} \sin \left (d x + c\right )^{3}}}{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)^4/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69746, size = 455, normalized size = 3.79 \begin{align*} -\frac{3 \, a^{2} b^{2} \sin \left (d x + c\right ) + 10 \, a^{3} b - 6 \, a b^{3} - 6 \,{\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 6 \,{\left (2 \, a^{2} b^{2} - b^{4} -{\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{6 \,{\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(3*a^2*b^2*sin(d*x + c) + 10*a^3*b - 6*a*b^3 - 6*(2*a^3*b - a*b^3)*cos(d*x + c)^2 + 6*(a^4 - 2*a^2*b^2 +
b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*log(b*sin(d*x + c) + a)*sin(d*x + c) + 6*(2*a^2*b^2 - b^4 - (2*a
^2*b^2 - b^4)*cos(d*x + c)^2)*log(1/2*sin(d*x + c))*sin(d*x + c))/((a^4*b*d*cos(d*x + c)^2 - a^4*b*d)*sin(d*x
+ c))

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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)**4/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.21953, size = 204, normalized size = 1.7 \begin{align*} \frac{\frac{6 \,{\left (2 \, a^{2} b - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} + \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^{4} b} - \frac{22 \, a^{2} b \sin \left (d x + c\right )^{3} - 11 \, b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right )^{2} - 3 \, a^{2} b \sin \left (d x + c\right ) + 2 \, a^{3}}{a^{4} \sin \left (d x + c\right )^{3}}}{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)^4/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*(2*a^2*b - b^3)*log(abs(sin(d*x + c)))/a^4 + 6*(a^4 - 2*a^2*b^2 + b^4)*log(abs(b*sin(d*x + c) + a))/(a^
4*b) - (22*a^2*b*sin(d*x + c)^3 - 11*b^3*sin(d*x + c)^3 - 12*a^3*sin(d*x + c)^2 + 6*a*b^2*sin(d*x + c)^2 - 3*a
^2*b*sin(d*x + c) + 2*a^3)/(a^4*sin(d*x + c)^3))/d