3.59 \(\int \frac{\csc ^4(c+d x)}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\left (a^2+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{b \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^4 d}+\frac{b \left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{a^4 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d} \]

[Out]

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

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Rubi [A]  time = 0.104453, antiderivative size = 108, 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 = {3516, 894} \[ -\frac{\left (a^2+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{b \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^4 d}+\frac{b \left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{a^4 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3516

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.455689, size = 95, normalized size = 0.88 \[ \frac{-2 \cot (c+d x) \left (a^3 \csc ^2(c+d x)+2 a^3+3 a b^2\right )-6 b \left (a^2+b^2\right ) (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+3 a^2 b \csc ^2(c+d x)}{6 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*b*Csc[c + d*x]^2 - 2*Cot[c + d*x]*(2*a^3 + 3*a*b^2 + a^3*Csc[c + d*x]^2) - 6*b*(a^2 + b^2)*(Log[Sin[c +
d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x]]))/(6*a^4*d)

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Maple [A]  time = 0.076, size = 144, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{ad\tan \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\tan \left ( dx+c \right ) }}+{\frac{b}{2\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}-{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}+{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^4/(a+b*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.06377, size = 131, normalized size = 1.21 \begin{align*} \frac{\frac{6 \,{\left (a^{2} b + b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4}} - \frac{6 \,{\left (a^{2} b + b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}} + \frac{3 \, a b \tan \left (d x + c\right ) - 6 \,{\left (a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{2} - 2 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(a^2*b + b^3)*log(b*tan(d*x + c) + a)/a^4 - 6*(a^2*b + b^3)*log(tan(d*x + c))/a^4 + (3*a*b*tan(d*x + c)
- 6*(a^2 + b^2)*tan(d*x + c)^2 - 2*a^2)/(a^3*tan(d*x + c)^3))/d

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Fricas [B]  time = 2.15412, size = 500, normalized size = 4.63 \begin{align*} -\frac{2 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, a^{2} b \sin \left (d x + c\right ) + 3 \,{\left (a^{2} b + b^{3} -{\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) \sin \left (d x + c\right ) - 3 \,{\left (a^{2} b + b^{3} -{\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) - 6 \,{\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )}{6 \,{\left (a^{4} d \cos \left (d x + c\right )^{2} - a^{4} 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*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.18105, size = 194, normalized size = 1.8 \begin{align*} -\frac{\frac{6 \,{\left (a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{6 \,{\left (a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b} - \frac{11 \, a^{2} b \tan \left (d x + c\right )^{3} + 11 \, b^{3} \tan \left (d x + c\right )^{3} - 6 \, a^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b \tan \left (d x + c\right ) - 2 \, a^{3}}{a^{4} \tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(a^2*b + b^3)*log(abs(tan(d*x + c)))/a^4 - 6*(a^2*b^2 + b^4)*log(abs(b*tan(d*x + c) + a))/(a^4*b) - (1
1*a^2*b*tan(d*x + c)^3 + 11*b^3*tan(d*x + c)^3 - 6*a^3*tan(d*x + c)^2 - 6*a*b^2*tan(d*x + c)^2 + 3*a^2*b*tan(d
*x + c) - 2*a^3)/(a^4*tan(d*x + c)^3))/d