3.29 \(\int \csc ^4(c+d x) (a+b \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=79 \[ -\frac{\left (a^2+b^2\right ) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0700101, antiderivative size = 79, 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} \[ -\frac{\left (a^2+b^2\right ) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 1.40886, size = 127, normalized size = 1.61 \[ -\frac{(a+b \tan (c+d x))^2 \left (\cos ^2(c+d x) \left (\left (2 a^2+3 b^2\right ) \cot (c+d x)+6 a b (\log (\cos (c+d x))-\log (\sin (c+d x)))\right )+a^2 \cot ^3(c+d x)+3 a b \cot ^2(c+d x)-\frac{3}{2} b^2 \sin (2 (c+d x))\right )}{3 d (a \cos (c+d x)+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.053, size = 104, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-2\,{\frac{{b}^{2}\cot \left ( dx+c \right ) }{d}}-{\frac{ab}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{ab\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{2\,{a}^{2}\cot \left ( dx+c \right ) }{3\,d}}-{\frac{{a}^{2}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.08606, size = 93, normalized size = 1.18 \begin{align*} \frac{6 \, a b \log \left (\tan \left (d x + c\right )\right ) + 3 \, b^{2} \tan \left (d x + c\right ) - \frac{3 \, a b \tan \left (d x + c\right ) + 3 \,{\left (a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.39439, size = 446, normalized size = 5.65 \begin{align*} -\frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 3 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \,{\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 3 \,{\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) + 3 \, b^{2}}{3 \,{\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.61879, size = 123, normalized size = 1.56 \begin{align*} \frac{6 \, a b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 3 \, b^{2} \tan \left (d x + c\right ) - \frac{11 \, a b \tan \left (d x + c\right )^{3} + 3 \, a^{2} \tan \left (d x + c\right )^{2} + 3 \, b^{2} \tan \left (d x + c\right )^{2} + 3 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, 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))^2,x, algorithm="giac")

[Out]

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