3.21 \(\int \csc ^6(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=87 \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d} \]

[Out]

-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/d - (2*a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[
c + d*x]^5)/(5*d) + (b*Log[Tan[c + d*x]])/d

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Rubi [A]  time = 0.107578, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {766} \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/d - (2*a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[
c + d*x]^5)/(5*d) + (b*Log[Tan[c + d*x]])/d

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.545438, size = 104, normalized size = 1.2 \[ -\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{b \left (\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.085, size = 94, normalized size = 1.1 \begin{align*} -{\frac{b}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{b}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{8\,\cot \left ( dx+c \right ) a}{15\,d}}-{\frac{\cot \left ( dx+c \right ) a \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{4\,\cot \left ( dx+c \right ) a \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{15\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13162, size = 97, normalized size = 1.11 \begin{align*} \frac{60 \, b \log \left (\tan \left (d x + c\right )\right ) - \frac{60 \, a \tan \left (d x + c\right )^{4} + 60 \, b \tan \left (d x + c\right )^{3} + 40 \, a \tan \left (d x + c\right )^{2} + 15 \, b \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b*log(tan(d*x + c)) - (60*a*tan(d*x + c)^4 + 60*b*tan(d*x + c)^3 + 40*a*tan(d*x + c)^2 + 15*b*tan(d*x
+ c) + 12*a)/tan(d*x + c)^5)/d

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Fricas [B]  time = 2.10032, size = 473, normalized size = 5.44 \begin{align*} -\frac{32 \, a \cos \left (d x + c\right )^{5} - 80 \, a \cos \left (d x + c\right )^{3} + 30 \,{\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 30 \,{\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) + 60 \, a \cos \left (d x + c\right ) - 15 \,{\left (2 \, b \cos \left (d x + c\right )^{2} - 3 \, b\right )} \sin \left (d x + c\right )}{60 \,{\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*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/60*(32*a*cos(d*x + c)^5 - 80*a*cos(d*x + c)^3 + 30*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + b)*log(cos(d*x
+ c)^2)*sin(d*x + c) - 30*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + b)*log(-1/4*cos(d*x + c)^2 + 1/4)*sin(d*x +
c) + 60*a*cos(d*x + c) - 15*(2*b*cos(d*x + c)^2 - 3*b)*sin(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*tan(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.38364, size = 113, normalized size = 1.3 \begin{align*} \frac{60 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{137 \, b \tan \left (d x + c\right )^{5} + 60 \, a \tan \left (d x + c\right )^{4} + 60 \, b \tan \left (d x + c\right )^{3} + 40 \, a \tan \left (d x + c\right )^{2} + 15 \, b \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b*log(abs(tan(d*x + c))) - (137*b*tan(d*x + c)^5 + 60*a*tan(d*x + c)^4 + 60*b*tan(d*x + c)^3 + 40*a*t
an(d*x + c)^2 + 15*b*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d