3.73 \(\int \csc ^6(c+d x) (a+b \sin ^2(c+d x)) \, dx\)

Optimal. Leaf size=65 \[ -\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]

[Out]

-((4*a + 5*b)*Cot[c + d*x])/(5*d) - ((4*a + 5*b)*Cot[c + d*x]^3)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d
)

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Rubi [A]  time = 0.042355, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 3767} \[ -\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6*(a + b*Sin[c + d*x]^2),x]

[Out]

-((4*a + 5*b)*Cot[c + d*x])/(5*d) - ((4*a + 5*b)*Cot[c + d*x]^3)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d
)

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}+\frac{1}{5} (4 a+5 b) \int \csc ^4(c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{(4 a+5 b) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{5 d}\\ &=-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.028352, size = 95, normalized size = 1.46 \[ -\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{2 b \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6*(a + b*Sin[c + d*x]^2),x]

[Out]

(-8*a*Cot[c + d*x])/(15*d) - (2*b*Cot[c + d*x])/(3*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (b*Cot[c +
d*x]*Csc[c + d*x]^2)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)

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Maple [A]  time = 0.053, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cot \left ( dx+c \right ) +b \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^6*(a+sin(d*x+c)^2*b),x)

[Out]

1/d*(a*(-8/15-1/5*csc(d*x+c)^4-4/15*csc(d*x+c)^2)*cot(d*x+c)+b*(-2/3-1/3*csc(d*x+c)^2)*cot(d*x+c))

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Maxima [A]  time = 0.943166, size = 61, normalized size = 0.94 \begin{align*} -\frac{15 \,{\left (a + b\right )} \tan \left (d x + c\right )^{4} + 5 \,{\left (2 \, a + b\right )} \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6*(a+b*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/15*(15*(a + b)*tan(d*x + c)^4 + 5*(2*a + b)*tan(d*x + c)^2 + 3*a)/(d*tan(d*x + c)^5)

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Fricas [A]  time = 1.61107, size = 208, normalized size = 3.2 \begin{align*} -\frac{2 \,{\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{5} - 5 \,{\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (a + b\right )} \cos \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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(csc(d*x+c)^6*(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

-1/15*(2*(4*a + 5*b)*cos(d*x + c)^5 - 5*(4*a + 5*b)*cos(d*x + c)^3 + 15*(a + b)*cos(d*x + c))/((d*cos(d*x + c)
^4 - 2*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(csc(d*x+c)**6*(a+b*sin(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.1379, size = 82, normalized size = 1.26 \begin{align*} -\frac{15 \, a \tan \left (d x + c\right )^{4} + 15 \, b \tan \left (d x + c\right )^{4} + 10 \, a \tan \left (d x + c\right )^{2} + 5 \, b \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6*(a+b*sin(d*x+c)^2),x, algorithm="giac")

[Out]

-1/15*(15*a*tan(d*x + c)^4 + 15*b*tan(d*x + c)^4 + 10*a*tan(d*x + c)^2 + 5*b*tan(d*x + c)^2 + 3*a)/(d*tan(d*x
+ c)^5)