3.665 \(\int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=118 \[ -\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0864115, antiderivative size = 118, 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 = {2836, 12, 88} \[ -\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 2836

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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.219729, size = 103, normalized size = 0.87 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{3 a \csc (c+d x)}{d}-\frac{a \left (\sin ^4(c+d x)-6 \sin ^2(c+d x)+2 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

(3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) + (3*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) - (a*(2*Csc[
c + d*x]^2 + 12*Log[Sin[c + d*x]] - 6*Sin[c + d*x]^2 + Sin[c + d*x]^4))/(4*d)

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Maple [A]  time = 0.068, size = 195, normalized size = 1.7 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{16\,a\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{3\,d}}+2\,{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{d}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x)

[Out]

-1/2/d*a/sin(d*x+c)^2*cos(d*x+c)^8-1/2*a*cos(d*x+c)^6/d-3/4*a*cos(d*x+c)^4/d-3/2*a*cos(d*x+c)^2/d-3*a*ln(sin(d
*x+c))/d-1/3/d*a/sin(d*x+c)^3*cos(d*x+c)^8+5/3/d*a/sin(d*x+c)*cos(d*x+c)^8+16/3*a*sin(d*x+c)/d+5/3/d*cos(d*x+c
)^6*sin(d*x+c)*a+2/d*cos(d*x+c)^4*sin(d*x+c)*a+8/3/d*cos(d*x+c)^2*sin(d*x+c)*a

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Maxima [A]  time = 1.02428, size = 124, normalized size = 1.05 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 36 \, a \log \left (\sin \left (d x + c\right )\right ) - 36 \, a \sin \left (d x + c\right ) - \frac{2 \,{\left (18 \, a \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(3*a*sin(d*x + c)^4 + 4*a*sin(d*x + c)^3 - 18*a*sin(d*x + c)^2 + 36*a*log(sin(d*x + c)) - 36*a*sin(d*x +
c) - 2*(18*a*sin(d*x + c)^2 - 3*a*sin(d*x + c) - 2*a)/sin(d*x + c)^3)/d

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Fricas [A]  time = 1.74723, size = 369, normalized size = 3.13 \begin{align*} -\frac{32 \, a \cos \left (d x + c\right )^{6} + 192 \, a \cos \left (d x + c\right )^{4} - 768 \, a \cos \left (d x + c\right )^{2} + 288 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \,{\left (8 \, a \cos \left (d x + c\right )^{6} + 24 \, a \cos \left (d x + c\right )^{4} - 51 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 512 \, a}{96 \,{\left (d \cos \left (d x + c\right )^{2} - 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)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/96*(32*a*cos(d*x + c)^6 + 192*a*cos(d*x + c)^4 - 768*a*cos(d*x + c)^2 + 288*(a*cos(d*x + c)^2 - a)*log(1/2*
sin(d*x + c))*sin(d*x + c) + 3*(8*a*cos(d*x + c)^6 + 24*a*cos(d*x + c)^4 - 51*a*cos(d*x + c)^2 + 3*a)*sin(d*x
+ c) + 512*a)/((d*cos(d*x + c)^2 - 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)**7*csc(d*x+c)**4*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.31783, size = 140, normalized size = 1.19 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 36 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 36 \, a \sin \left (d x + c\right ) - \frac{2 \,{\left (33 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/12*(3*a*sin(d*x + c)^4 + 4*a*sin(d*x + c)^3 - 18*a*sin(d*x + c)^2 + 36*a*log(abs(sin(d*x + c))) - 36*a*sin(
d*x + c) - 2*(33*a*sin(d*x + c)^3 + 18*a*sin(d*x + c)^2 - 3*a*sin(d*x + c) - 2*a)/sin(d*x + c)^3)/d