3.72 \(\int \csc ^4(c+d x) (a+b \sin ^2(c+d x)) \, dx\)

Optimal. Leaf size=43 \[ -\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0357116, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3012, 3767, 8} \[ -\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0264045, size = 49, normalized size = 1.14 \[ -\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \cot (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]

[Out]

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

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Maple [A]  time = 0.052, size = 35, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) -b\cot \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^4*(a+sin(d*x+c)^2*b),x)

[Out]

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

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Maxima [A]  time = 0.950237, size = 38, normalized size = 0.88 \begin{align*} -\frac{3 \,{\left (a + b\right )} \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58397, size = 132, normalized size = 3.07 \begin{align*} -\frac{{\left (2 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (a + b\right )} \cos \left (d x + c\right )}{3 \,{\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(csc(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.15288, size = 50, normalized size = 1.16 \begin{align*} -\frac{3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm="giac")

[Out]

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