3.43 \(\int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=67 \[ \frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d} \]

[Out]

(4*a^3*Log[1 - Cos[c + d*x]])/d - (4*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/
(2*d)

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Rubi [A]  time = 0.125251, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2836, 12, 88} \[ \frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*Log[1 - Cos[c + d*x]])/d - (4*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/
(2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

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

Mathematica [A]  time = 0.131181, size = 81, normalized size = 1.21 \[ \frac{a^3 \sec ^2(c+d x) \left (6 \cos (c+d x)+8 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-4 \log (\cos (c+d x))-4 \cos (2 (c+d x)) \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )+1\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + 6*Cos[c + d*x] - 4*Log[Cos[c + d*x]] - 4*Cos[2*(c + d*x)]*(Log[Cos[c + d*x]] - 2*Log[Sin[(c + d*x)/2
]]) + 8*Log[Sin[(c + d*x)/2]])*Sec[c + d*x]^2)/(2*d)

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Maple [A]  time = 0.045, size = 49, normalized size = 0.7 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*sec(d*x+c)^2/d+3*a^3*sec(d*x+c)/d+4/d*a^3*ln(-1+sec(d*x+c))

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Maxima [A]  time = 0.98626, size = 76, normalized size = 1.13 \begin{align*} \frac{8 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 8 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(8*a^3*log(cos(d*x + c) - 1) - 8*a^3*log(cos(d*x + c)) + (6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d

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Fricas [A]  time = 1.79563, size = 197, normalized size = 2.94 \begin{align*} -\frac{8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(8*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) - 8*a^3*cos(d*x + c)^2*log(-1/2*cos(d*x + c) + 1/2) - 6*a^3*cos(
d*x + c) - a^3)/(d*cos(d*x + c)^2)

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

[Out]

a**3*(Integral(3*csc(c + d*x)*sec(c + d*x), x) + Integral(3*csc(c + d*x)*sec(c + d*x)**2, x) + Integral(csc(c
+ d*x)*sec(c + d*x)**3, x) + Integral(csc(c + d*x), x))

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Giac [B]  time = 1.27331, size = 192, normalized size = 2.87 \begin{align*} \frac{2 \,{\left (2 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{6 \, a^{3} + \frac{8 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 2*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) +
1) - 1)) + (6*a^3 + 8*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2)/d