3.40 \(\int \csc ^6(c+d x) (a+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=167 \[ -\frac{a \left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{b \left (6 a^2+b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{a \left (a^2+6 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2+2 b^2\right ) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^4(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d} \]

[Out]

-((a*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (b*(6*a^2 + b^2)*Cot[c + d*x]^2)/(2*d) - (a*(2*a^2 + 3*b^2)*Cot[c + d*x]
^3)/(3*d) - (3*a^2*b*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Log[Tan[c + d*x]]
)/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.134132, antiderivative size = 167, 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, 948} \[ -\frac{a \left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{b \left (6 a^2+b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{a \left (a^2+6 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2+2 b^2\right ) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^4(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]

[Out]

-((a*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (b*(6*a^2 + b^2)*Cot[c + d*x]^2)/(2*d) - (a*(2*a^2 + 3*b^2)*Cot[c + d*x]
^3)/(3*d) - (3*a^2*b*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Log[Tan[c + d*x]]
)/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*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 948

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] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rubi steps

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

Mathematica [B]  time = 1.75822, size = 515, normalized size = 3.08 \[ -\frac{\csc ^5(c+d x) \sec ^2(c+d x) \left (40 a \left (5 a^2+3 b^2\right ) \cos (c+d x)+8 \left (a^3+15 a b^2\right ) \cos (3 (c+d x))+360 a^2 b \sin (c+d x)+270 a^2 b \sin (3 (c+d x))-90 a^2 b \sin (5 (c+d x))-225 a^2 b \sin (c+d x) \log (\sin (c+d x))-45 a^2 b \sin (3 (c+d x)) \log (\sin (c+d x))+135 a^2 b \sin (5 (c+d x)) \log (\sin (c+d x))-45 a^2 b \sin (7 (c+d x)) \log (\sin (c+d x))+225 a^2 b \sin (c+d x) \log (\cos (c+d x))+45 a^2 b \sin (3 (c+d x)) \log (\cos (c+d x))-135 a^2 b \sin (5 (c+d x)) \log (\cos (c+d x))+45 a^2 b \sin (7 (c+d x)) \log (\cos (c+d x))-24 a^3 \cos (5 (c+d x))+8 a^3 \cos (7 (c+d x))-360 a b^2 \cos (5 (c+d x))+120 a b^2 \cos (7 (c+d x))-240 b^3 \sin (c+d x)+180 b^3 \sin (3 (c+d x))-60 b^3 \sin (5 (c+d x))-150 b^3 \sin (c+d x) \log (\sin (c+d x))-30 b^3 \sin (3 (c+d x)) \log (\sin (c+d x))+90 b^3 \sin (5 (c+d x)) \log (\sin (c+d x))-30 b^3 \sin (7 (c+d x)) \log (\sin (c+d x))+150 b^3 \sin (c+d x) \log (\cos (c+d x))+30 b^3 \sin (3 (c+d x)) \log (\cos (c+d x))-90 b^3 \sin (5 (c+d x)) \log (\cos (c+d x))+30 b^3 \sin (7 (c+d x)) \log (\cos (c+d x))\right )}{960 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]

[Out]

-(Csc[c + d*x]^5*Sec[c + d*x]^2*(40*a*(5*a^2 + 3*b^2)*Cos[c + d*x] + 8*(a^3 + 15*a*b^2)*Cos[3*(c + d*x)] - 24*
a^3*Cos[5*(c + d*x)] - 360*a*b^2*Cos[5*(c + d*x)] + 8*a^3*Cos[7*(c + d*x)] + 120*a*b^2*Cos[7*(c + d*x)] + 360*
a^2*b*Sin[c + d*x] - 240*b^3*Sin[c + d*x] + 225*a^2*b*Log[Cos[c + d*x]]*Sin[c + d*x] + 150*b^3*Log[Cos[c + d*x
]]*Sin[c + d*x] - 225*a^2*b*Log[Sin[c + d*x]]*Sin[c + d*x] - 150*b^3*Log[Sin[c + d*x]]*Sin[c + d*x] + 270*a^2*
b*Sin[3*(c + d*x)] + 180*b^3*Sin[3*(c + d*x)] + 45*a^2*b*Log[Cos[c + d*x]]*Sin[3*(c + d*x)] + 30*b^3*Log[Cos[c
+ d*x]]*Sin[3*(c + d*x)] - 45*a^2*b*Log[Sin[c + d*x]]*Sin[3*(c + d*x)] - 30*b^3*Log[Sin[c + d*x]]*Sin[3*(c +
d*x)] - 90*a^2*b*Sin[5*(c + d*x)] - 60*b^3*Sin[5*(c + d*x)] - 135*a^2*b*Log[Cos[c + d*x]]*Sin[5*(c + d*x)] - 9
0*b^3*Log[Cos[c + d*x]]*Sin[5*(c + d*x)] + 135*a^2*b*Log[Sin[c + d*x]]*Sin[5*(c + d*x)] + 90*b^3*Log[Sin[c + d
*x]]*Sin[5*(c + d*x)] + 45*a^2*b*Log[Cos[c + d*x]]*Sin[7*(c + d*x)] + 30*b^3*Log[Cos[c + d*x]]*Sin[7*(c + d*x)
] - 45*a^2*b*Log[Sin[c + d*x]]*Sin[7*(c + d*x)] - 30*b^3*Log[Sin[c + d*x]]*Sin[7*(c + d*x)]))/(960*d)

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Maple [A]  time = 0.069, size = 230, normalized size = 1.4 \begin{align*}{\frac{{b}^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{a{b}^{2}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) }}+4\,{\frac{a{b}^{2}}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-8\,{\frac{a{b}^{2}\cot \left ( dx+c \right ) }{d}}-{\frac{3\,b{a}^{2}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,b{a}^{2}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b{a}^{2}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{8\,{a}^{3}\cot \left ( dx+c \right ) }{15\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{4\,{a}^{3}\cot \left ( dx+c \right ) \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))^3,x)

[Out]

1/2/d*b^3/sin(d*x+c)^2/cos(d*x+c)^2-1/d*b^3/sin(d*x+c)^2+2/d*b^3*ln(tan(d*x+c))-1/d*a*b^2/sin(d*x+c)^3/cos(d*x
+c)+4/d*a*b^2/sin(d*x+c)/cos(d*x+c)-8/d*a*b^2*cot(d*x+c)-3/4/d*b*a^2/sin(d*x+c)^4-3/2/d*b*a^2/sin(d*x+c)^2+3*a
^2*b*ln(tan(d*x+c))/d-8/15*a^3*cot(d*x+c)/d-1/5/d*a^3*cot(d*x+c)*csc(d*x+c)^4-4/15/d*a^3*cot(d*x+c)*csc(d*x+c)
^2

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Maxima [A]  time = 1.11848, size = 192, normalized size = 1.15 \begin{align*} \frac{30 \, b^{3} \tan \left (d x + c\right )^{2} + 180 \, a b^{2} \tan \left (d x + c\right ) + 60 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{60 \,{\left (a^{3} + 6 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} + 45 \, a^{2} b \tan \left (d x + c\right ) + 30 \,{\left (6 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{3} + 12 \, a^{3} + 20 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\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))^3,x, algorithm="maxima")

[Out]

1/60*(30*b^3*tan(d*x + c)^2 + 180*a*b^2*tan(d*x + c) + 60*(3*a^2*b + 2*b^3)*log(tan(d*x + c)) - (60*(a^3 + 6*a
*b^2)*tan(d*x + c)^4 + 45*a^2*b*tan(d*x + c) + 30*(6*a^2*b + b^3)*tan(d*x + c)^3 + 12*a^3 + 20*(2*a^3 + 3*a*b^
2)*tan(d*x + c)^2)/tan(d*x + c)^5)/d

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Fricas [B]  time = 2.28379, size = 834, normalized size = 4.99 \begin{align*} -\frac{32 \,{\left (a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 80 \,{\left (a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 180 \, a b^{2} \cos \left (d x + c\right ) + 60 \,{\left (a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 30 \,{\left ({\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 30 \,{\left ({\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (3 \, a^{2} b + 2 \, 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 ) - 15 \,{\left (2 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 2 \, b^{3} - 3 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \,{\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\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))^3,x, algorithm="fricas")

[Out]

-1/60*(32*(a^3 + 15*a*b^2)*cos(d*x + c)^7 - 80*(a^3 + 15*a*b^2)*cos(d*x + c)^5 - 180*a*b^2*cos(d*x + c) + 60*(
a^3 + 15*a*b^2)*cos(d*x + c)^3 + 30*((3*a^2*b + 2*b^3)*cos(d*x + c)^6 - 2*(3*a^2*b + 2*b^3)*cos(d*x + c)^4 + (
3*a^2*b + 2*b^3)*cos(d*x + c)^2)*log(cos(d*x + c)^2)*sin(d*x + c) - 30*((3*a^2*b + 2*b^3)*cos(d*x + c)^6 - 2*(
3*a^2*b + 2*b^3)*cos(d*x + c)^4 + (3*a^2*b + 2*b^3)*cos(d*x + c)^2)*log(-1/4*cos(d*x + c)^2 + 1/4)*sin(d*x + c
) - 15*(2*(3*a^2*b + 2*b^3)*cos(d*x + c)^4 + 2*b^3 - 3*(3*a^2*b + 2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((d*cos
(d*x + c)^6 - 2*d*cos(d*x + c)^4 + d*cos(d*x + c)^2)*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 2.03189, size = 255, normalized size = 1.53 \begin{align*} \frac{30 \, b^{3} \tan \left (d x + c\right )^{2} + 180 \, a b^{2} \tan \left (d x + c\right ) + 60 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{411 \, a^{2} b \tan \left (d x + c\right )^{5} + 274 \, b^{3} \tan \left (d x + c\right )^{5} + 60 \, a^{3} \tan \left (d x + c\right )^{4} + 360 \, a b^{2} \tan \left (d x + c\right )^{4} + 180 \, a^{2} b \tan \left (d x + c\right )^{3} + 30 \, b^{3} \tan \left (d x + c\right )^{3} + 40 \, a^{3} \tan \left (d x + c\right )^{2} + 60 \, a b^{2} \tan \left (d x + c\right )^{2} + 45 \, a^{2} b \tan \left (d x + c\right ) + 12 \, a^{3}}{\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))^3,x, algorithm="giac")

[Out]

1/60*(30*b^3*tan(d*x + c)^2 + 180*a*b^2*tan(d*x + c) + 60*(3*a^2*b + 2*b^3)*log(abs(tan(d*x + c))) - (411*a^2*
b*tan(d*x + c)^5 + 274*b^3*tan(d*x + c)^5 + 60*a^3*tan(d*x + c)^4 + 360*a*b^2*tan(d*x + c)^4 + 180*a^2*b*tan(d
*x + c)^3 + 30*b^3*tan(d*x + c)^3 + 40*a^3*tan(d*x + c)^2 + 60*a*b^2*tan(d*x + c)^2 + 45*a^2*b*tan(d*x + c) +
12*a^3)/tan(d*x + c)^5)/d