3.75 \(\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3 b}{a^4 d (a+b \tan (c+d x))}-\frac{b}{a^3 d (a+b \tan (c+d x))^2}-\frac{b}{3 a^2 d (a+b \tan (c+d x))^3}-\frac{4 b \log (\tan (c+d x))}{a^5 d}+\frac{4 b \log (a+b \tan (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a^4 d} \]

[Out]

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

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Rubi [A]  time = 0.0884231, antiderivative size = 116, 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, 44} \[ -\frac{3 b}{a^4 d (a+b \tan (c+d x))}-\frac{b}{a^3 d (a+b \tan (c+d x))^2}-\frac{b}{3 a^2 d (a+b \tan (c+d x))^3}-\frac{4 b \log (\tan (c+d x))}{a^5 d}+\frac{4 b \log (a+b \tan (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 2.06357, size = 259, normalized size = 2.23 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac{a^2 b^4 \tan (c+d x)}{a^2+b^2}-\frac{2 a^2 b^3 \left (3 a^2+2 b^2\right ) (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}+\frac{b^2 \left (23 a^2 b^2+18 a^4+9 b^4\right ) \tan (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (a^2+b^2\right )^2}-3 a \sin ^2(c+d x) (a \cot (c+d x)+b)^3-12 b \cos ^2(c+d x) \log (\sin (c+d x)) (a+b \tan (c+d x))^3+12 b \cos ^2(c+d x) (a+b \tan (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))\right )}{3 a^5 d (a+b \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-3*a*(b + a*Cot[c + d*x])^3*Sin[c + d*x]^2 + (a^2*b^4*Tan[c
+ d*x])/(a^2 + b^2) + (b^2*(18*a^4 + 23*a^2*b^2 + 9*b^4)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*Tan[c + d*x])/(a
^2 + b^2)^2 - (2*a^2*b^3*(3*a^2 + 2*b^2)*(a + b*Tan[c + d*x]))/(a^2 + b^2)^2 - 12*b*Cos[c + d*x]^2*Log[Sin[c +
d*x]]*(a + b*Tan[c + d*x])^3 + 12*b*Cos[c + d*x]^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a + b*Tan[c + d*x])^
3))/(3*a^5*d*(a + b*Tan[c + d*x])^4)

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Maple [A]  time = 0.131, size = 117, normalized size = 1. \begin{align*} -{\frac{1}{d{a}^{4}\tan \left ( dx+c \right ) }}-4\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{5}d}}-{\frac{b}{3\,{a}^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}+4\,{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{a}^{5}d}}-3\,{\frac{b}{d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{b}{{a}^{3}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/a^4/tan(d*x+c)-4*b*ln(tan(d*x+c))/a^5/d-1/3*b/a^2/d/(a+b*tan(d*x+c))^3+4*b*ln(a+b*tan(d*x+c))/a^5/d-3*b/a
^4/d/(a+b*tan(d*x+c))-b/a^3/d/(a+b*tan(d*x+c))^2

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Maxima [A]  time = 1.14022, size = 189, normalized size = 1.63 \begin{align*} -\frac{\frac{12 \, b^{3} \tan \left (d x + c\right )^{3} + 30 \, a b^{2} \tan \left (d x + c\right )^{2} + 22 \, a^{2} b \tan \left (d x + c\right ) + 3 \, a^{3}}{a^{4} b^{3} \tan \left (d x + c\right )^{4} + 3 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 3 \, a^{6} b \tan \left (d x + c\right )^{2} + a^{7} \tan \left (d x + c\right )} - \frac{12 \, b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5}} + \frac{12 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((12*b^3*tan(d*x + c)^3 + 30*a*b^2*tan(d*x + c)^2 + 22*a^2*b*tan(d*x + c) + 3*a^3)/(a^4*b^3*tan(d*x + c)^
4 + 3*a^5*b^2*tan(d*x + c)^3 + 3*a^6*b*tan(d*x + c)^2 + a^7*tan(d*x + c)) - 12*b*log(b*tan(d*x + c) + a)/a^5 +
12*b*log(tan(d*x + c))/a^5)/d

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Fricas [B]  time = 2.84577, size = 1901, normalized size = 16.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*(13*a^6*b^4 + 15*a^4*b^6 + 6*a^2*b^8 - (3*a^10 + 18*a^8*b^2 - 49*a^6*b^4 - 84*a^4*b^6 - 36*a^2*b^8)*cos(d
*x + c)^4 + (9*a^8*b^2 - 71*a^6*b^4 - 102*a^4*b^6 - 42*a^2*b^8)*cos(d*x + c)^2 + 6*(a^6*b^4 + 3*a^4*b^6 + 3*a^
2*b^8 + b^10 - (3*a^8*b^2 + 8*a^6*b^4 + 6*a^4*b^6 - b^10)*cos(d*x + c)^4 + (3*a^8*b^2 + 7*a^6*b^4 + 3*a^4*b^6
- 3*a^2*b^8 - 2*b^10)*cos(d*x + c)^2 + ((a^9*b - 6*a^5*b^5 - 8*a^3*b^7 - 3*a*b^9)*cos(d*x + c)^3 + 3*(a^7*b^3
+ 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*cos(d*x + c))*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)
*cos(d*x + c)^2 + b^2) - 6*(a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10 - (3*a^8*b^2 + 8*a^6*b^4 + 6*a^4*b^6 - b^10
)*cos(d*x + c)^4 + (3*a^8*b^2 + 7*a^6*b^4 + 3*a^4*b^6 - 3*a^2*b^8 - 2*b^10)*cos(d*x + c)^2 + ((a^9*b - 6*a^5*b
^5 - 8*a^3*b^7 - 3*a*b^9)*cos(d*x + c)^3 + 3*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*cos(d*x + c))*sin(d*x +
c))*log(-1/4*cos(d*x + c)^2 + 1/4) - ((9*a^9*b + 78*a^7*b^3 + 69*a^5*b^5 + 4*a^3*b^7 - 12*a*b^9)*cos(d*x + c)
^3 - 3*(9*a^7*b^3 + 3*a^5*b^5 - 6*a^3*b^7 - 4*a*b^9)*cos(d*x + c))*sin(d*x + c))/((3*a^13*b + 8*a^11*b^3 + 6*a
^9*b^5 - a^5*b^9)*d*cos(d*x + c)^4 - (3*a^13*b + 7*a^11*b^3 + 3*a^9*b^5 - 3*a^7*b^7 - 2*a^5*b^9)*d*cos(d*x + c
)^2 - (a^11*b^3 + 3*a^9*b^5 + 3*a^7*b^7 + a^5*b^9)*d - ((a^14 - 6*a^10*b^4 - 8*a^8*b^6 - 3*a^6*b^8)*d*cos(d*x
+ c)^3 + 3*(a^12*b^2 + 3*a^10*b^4 + 3*a^8*b^6 + a^6*b^8)*d*cos(d*x + c))*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)**2/(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.2252, size = 174, normalized size = 1.5 \begin{align*} \frac{\frac{12 \, b \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5}} - \frac{12 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} + \frac{3 \,{\left (4 \, b \tan \left (d x + c\right ) - a\right )}}{a^{5} \tan \left (d x + c\right )} - \frac{22 \, b^{4} \tan \left (d x + c\right )^{3} + 75 \, a b^{3} \tan \left (d x + c\right )^{2} + 87 \, a^{2} b^{2} \tan \left (d x + c\right ) + 35 \, a^{3} b}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} a^{5}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(12*b*log(abs(b*tan(d*x + c) + a))/a^5 - 12*b*log(abs(tan(d*x + c)))/a^5 + 3*(4*b*tan(d*x + c) - a)/(a^5*t
an(d*x + c)) - (22*b^4*tan(d*x + c)^3 + 75*a*b^3*tan(d*x + c)^2 + 87*a^2*b^2*tan(d*x + c) + 35*a^3*b)/((b*tan(
d*x + c) + a)^3*a^5))/d