3.1207 \(\int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=81 \[ -\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0529924, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2721, 766} \[ -\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2721

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.243567, size = 87, normalized size = 1.07 \[ \frac{a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 136, normalized size = 1.7 \begin{align*} -{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}+{\frac{8\,b\sin \left ( dx+c \right ) }{3\,d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}b}{d}}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) b}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995989, size = 93, normalized size = 1.15 \begin{align*} \frac{12 \, a \log \left (\sin \left (d x + c\right )\right ) + 12 \, b \sin \left (d x + c\right ) + \frac{24 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 4 \, b \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*a*log(sin(d*x + c)) + 12*b*sin(d*x + c) + (24*b*sin(d*x + c)^3 + 12*a*sin(d*x + c)^2 - 4*b*sin(d*x +
c) - 3*a)/sin(d*x + c)^4)/d

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Fricas [A]  time = 1.70965, size = 292, normalized size = 3.6 \begin{align*} -\frac{12 \, a \cos \left (d x + c\right )^{2} - 12 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 4 \,{\left (3 \, b \cos \left (d x + c\right )^{4} - 12 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) - 9 \, a}{12 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*a*cos(d*x + c)^2 - 12*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*sin(d*x + c)) - 4*(3*b*cos
(d*x + c)^4 - 12*b*cos(d*x + c)^2 + 8*b)*sin(d*x + c) - 9*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + 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)**5*(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.19303, size = 111, normalized size = 1.37 \begin{align*} \frac{12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, b \sin \left (d x + c\right ) - \frac{25 \, a \sin \left (d x + c\right )^{4} - 24 \, b \sin \left (d x + c\right )^{3} - 12 \, a \sin \left (d x + c\right )^{2} + 4 \, b \sin \left (d x + c\right ) + 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*a*log(abs(sin(d*x + c))) + 12*b*sin(d*x + c) - (25*a*sin(d*x + c)^4 - 24*b*sin(d*x + c)^3 - 12*a*sin(
d*x + c)^2 + 4*b*sin(d*x + c) + 3*a)/sin(d*x + c)^4)/d