3.1 \(\int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx\)

Optimal. Leaf size=91 \[ -\frac{2 a A \cot ^3(c+d x)}{3 d}-\frac{2 a A \cot (c+d x)}{d}-\frac{7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]

[Out]

(-7*a*A*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a*A*Cot[c + d*x])/d - (2*a*A*Cot[c + d*x]^3)/(3*d) - (7*a*A*Cot[c +
d*x]*Csc[c + d*x])/(8*d) - (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)

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Rubi [A]  time = 0.102302, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {21, 3788, 3767, 4046, 3768, 3770} \[ -\frac{2 a A \cot ^3(c+d x)}{3 d}-\frac{2 a A \cot (c+d x)}{d}-\frac{7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^3*(a + a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

(-7*a*A*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a*A*Cot[c + d*x])/d - (2*a*A*Cot[c + d*x]^3)/(3*d) - (7*a*A*Cot[c +
d*x]*Csc[c + d*x])/(8*d) - (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3788

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Dist[(2*a*b)/
d, Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b
, d, e, f, n}, x]

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 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx &=\frac{A \int \csc ^3(c+d x) (a+a \csc (c+d x))^2 \, dx}{a}\\ &=\frac{A \int \csc ^3(c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}+(2 a A) \int \csc ^4(c+d x) \, dx\\ &=-\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{4} (7 a A) \int \csc ^3(c+d x) \, dx-\frac{(2 a A) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{2 a A \cot (c+d x)}{d}-\frac{2 a A \cot ^3(c+d x)}{3 d}-\frac{7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{8} (7 a A) \int \csc (c+d x) \, dx\\ &=-\frac{7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{2 a A \cot (c+d x)}{d}-\frac{2 a A \cot ^3(c+d x)}{3 d}-\frac{7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0568929, size = 163, normalized size = 1.79 \[ -\frac{4 a A \cot (c+d x)}{3 d}-\frac{a A \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{7 a A \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a A \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{7 a A \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{7 a A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{7 a A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{2 a A \cot (c+d x) \csc ^2(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^3*(a + a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

(-4*a*A*Cot[c + d*x])/(3*d) - (7*a*A*Csc[(c + d*x)/2]^2)/(32*d) - (a*A*Csc[(c + d*x)/2]^4)/(64*d) - (2*a*A*Cot
[c + d*x]*Csc[c + d*x]^2)/(3*d) - (7*a*A*Log[Cos[(c + d*x)/2]])/(8*d) + (7*a*A*Log[Sin[(c + d*x)/2]])/(8*d) +
(7*a*A*Sec[(c + d*x)/2]^2)/(32*d) + (a*A*Sec[(c + d*x)/2]^4)/(64*d)

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Maple [A]  time = 0.051, size = 99, normalized size = 1.1 \begin{align*} -{\frac{7\,Aa\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{8\,d}}+{\frac{7\,Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{4\,Aa\cot \left ( dx+c \right ) }{3\,d}}-{\frac{2\,Aa\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{Aa\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3*(a+a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

-7/8*a*A*cot(d*x+c)*csc(d*x+c)/d+7/8/d*A*a*ln(csc(d*x+c)-cot(d*x+c))-4/3*a*A*cot(d*x+c)/d-2/3*a*A*cot(d*x+c)*c
sc(d*x+c)^2/d-1/4*a*A*cot(d*x+c)*csc(d*x+c)^3/d

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Maxima [A]  time = 0.984284, size = 196, normalized size = 2.15 \begin{align*} \frac{3 \, A a{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, A a{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{32 \,{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*(a+a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="maxima")

[Out]

1/48*(3*A*a*(2*(3*cos(d*x + c)^3 - 5*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c
) + 1) + 3*log(cos(d*x + c) - 1)) + 12*A*a*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - log(cos(d*x + c) + 1) + log(
cos(d*x + c) - 1)) - 32*(3*tan(d*x + c)^2 + 1)*A*a/tan(d*x + c)^3)/d

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Fricas [A]  time = 0.501738, size = 439, normalized size = 4.82 \begin{align*} \frac{42 \, A a \cos \left (d x + c\right )^{3} - 54 \, A a \cos \left (d x + c\right ) - 21 \,{\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 21 \,{\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 32 \,{\left (2 \, A a \cos \left (d x + c\right )^{3} - 3 \, A a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \,{\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(csc(d*x+c)^3*(a+a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(42*A*a*cos(d*x + c)^3 - 54*A*a*cos(d*x + c) - 21*(A*a*cos(d*x + c)^4 - 2*A*a*cos(d*x + c)^2 + A*a)*log(1
/2*cos(d*x + c) + 1/2) + 21*(A*a*cos(d*x + c)^4 - 2*A*a*cos(d*x + c)^2 + A*a)*log(-1/2*cos(d*x + c) + 1/2) + 3
2*(2*A*a*cos(d*x + c)^3 - 3*A*a*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} A a \left (\int \csc ^{3}{\left (c + d x \right )}\, dx + \int 2 \csc ^{4}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**3*(a+a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

A*a*(Integral(csc(c + d*x)**3, x) + Integral(2*csc(c + d*x)**4, x) + Integral(csc(c + d*x)**5, x))

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Giac [A]  time = 1.44568, size = 209, normalized size = 2.3 \begin{align*} \frac{3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 168 \, A a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 144 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{350 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 144 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*(a+a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="giac")

[Out]

1/192*(3*A*a*tan(1/2*d*x + 1/2*c)^4 + 16*A*a*tan(1/2*d*x + 1/2*c)^3 + 48*A*a*tan(1/2*d*x + 1/2*c)^2 + 168*A*a*
log(abs(tan(1/2*d*x + 1/2*c))) + 144*A*a*tan(1/2*d*x + 1/2*c) - (350*A*a*tan(1/2*d*x + 1/2*c)^4 + 144*A*a*tan(
1/2*d*x + 1/2*c)^3 + 48*A*a*tan(1/2*d*x + 1/2*c)^2 + 16*A*a*tan(1/2*d*x + 1/2*c) + 3*A*a)/tan(1/2*d*x + 1/2*c)
^4)/d