3.38 \(\int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=113 \[ -\frac{a \left (a^2+3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d} \]

[Out]

-((a*(a^2 + 3*b^2)*Cot[c + d*x])/d) - (3*a^2*b*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d) + (b*(3*a^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 = 0.0860254, antiderivative size = 113, 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} \[ -\frac{a \left (a^2+3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

-((a*(a^2 + 3*b^2)*Cot[c + d*x])/d) - (3*a^2*b*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d) + (b*(3*a^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 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))^3 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^3 \left (b^2+x^2\right )}{x^4} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (3 a+\frac{a^3 b^2}{x^4}+\frac{3 a^2 b^2}{x^3}+\frac{a^3+3 a b^2}{x^2}+\frac{3 a^2+b^2}{x}+x\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{a \left (a^2+3 b^2\right ) \cot (c+d x)}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{b \left (3 a^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 [A]  time = 2.20723, size = 212, normalized size = 1.88 \[ \frac{\sec ^2(c+d x) (a \cot (c+d x)+b)^3 \left (-2 \sin (c+d x) \left (6 \left (3 a^2 b+b^3\right ) \cos (2 (c+d x))-3 b \left (3 a^2+b^2\right ) \cos (4 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))-9 a^2 b \log (\sin (c+d x))+9 a^2 b \log (\cos (c+d x))+18 a^2 b+2 a^3 \sin (4 (c+d x))+18 a b^2 \sin (4 (c+d x))-3 b^3 \log (\sin (c+d x))+3 b^3 \log (\cos (c+d x))-6 b^3\right )-16 a^3 \cos (c+d x)\right )}{48 d (a \cos (c+d x)+b \sin (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

((b + a*Cot[c + d*x])^3*Sec[c + d*x]^2*(-16*a^3*Cos[c + d*x] - 2*Sin[c + d*x]*(18*a^2*b - 6*b^3 + 6*(3*a^2*b +
b^3)*Cos[2*(c + d*x)] + 9*a^2*b*Log[Cos[c + d*x]] + 3*b^3*Log[Cos[c + d*x]] - 3*b*(3*a^2 + b^2)*Cos[4*(c + d*
x)]*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]) - 9*a^2*b*Log[Sin[c + d*x]] - 3*b^3*Log[Sin[c + d*x]] + 2*a^3*Sin[
4*(c + d*x)] + 18*a*b^2*Sin[4*(c + d*x)])))/(48*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)

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Maple [A]  time = 0.068, size = 141, normalized size = 1.3 \begin{align*}{\frac{{b}^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{a{b}^{2}}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-6\,{\frac{a{b}^{2}\cot \left ( dx+c \right ) }{d}}-{\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{2\,{a}^{3}\cot \left ( dx+c \right ) }{3\,d}}-{\frac{{a}^{3}\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))^3,x)

[Out]

1/2/d*b^3/cos(d*x+c)^2+1/d*b^3*ln(tan(d*x+c))+3/d*a*b^2/sin(d*x+c)/cos(d*x+c)-6/d*a*b^2*cot(d*x+c)-3/2/d*b*a^2
/sin(d*x+c)^2+3*a^2*b*ln(tan(d*x+c))/d-2/3*a^3*cot(d*x+c)/d-1/3/d*a^3*cot(d*x+c)*csc(d*x+c)^2

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Maxima [A]  time = 1.0644, size = 132, normalized size = 1.17 \begin{align*} \frac{3 \, b^{3} \tan \left (d x + c\right )^{2} + 18 \, a b^{2} \tan \left (d x + c\right ) + 6 \,{\left (3 \, a^{2} b + b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{9 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3} + 6 \,{\left (a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\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))^3,x, algorithm="maxima")

[Out]

1/6*(3*b^3*tan(d*x + c)^2 + 18*a*b^2*tan(d*x + c) + 6*(3*a^2*b + b^3)*log(tan(d*x + c)) - (9*a^2*b*tan(d*x + c
) + 2*a^3 + 6*(a^3 + 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^3)/d

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Fricas [B]  time = 2.01836, size = 575, normalized size = 5.09 \begin{align*} -\frac{4 \,{\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 18 \, a b^{2} \cos \left (d x + c\right ) - 6 \,{\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left ({\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} -{\left (3 \, a^{2} b + 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 ) - 3 \,{\left ({\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} -{\left (3 \, 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 ) + 3 \,{\left (b^{3} -{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \,{\left (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)^4*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/6*(4*(a^3 + 9*a*b^2)*cos(d*x + c)^5 + 18*a*b^2*cos(d*x + c) - 6*(a^3 + 9*a*b^2)*cos(d*x + c)^3 + 3*((3*a^2*
b + b^3)*cos(d*x + c)^4 - (3*a^2*b + b^3)*cos(d*x + c)^2)*log(cos(d*x + c)^2)*sin(d*x + c) - 3*((3*a^2*b + b^3
)*cos(d*x + c)^4 - (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) + 3*(b^3 - (3*a
^2*b + b^3)*cos(d*x + c)^2)*sin(d*x + c))/((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)**4*(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 2.04811, size = 180, normalized size = 1.59 \begin{align*} \frac{3 \, b^{3} \tan \left (d x + c\right )^{2} + 18 \, a b^{2} \tan \left (d x + c\right ) + 6 \,{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{33 \, 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} + 18 \, a b^{2} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b \tan \left (d x + c\right ) + 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))^3,x, algorithm="giac")

[Out]

1/6*(3*b^3*tan(d*x + c)^2 + 18*a*b^2*tan(d*x + c) + 6*(3*a^2*b + b^3)*log(abs(tan(d*x + c))) - (33*a^2*b*tan(d
*x + c)^3 + 11*b^3*tan(d*x + c)^3 + 6*a^3*tan(d*x + c)^2 + 18*a*b^2*tan(d*x + c)^2 + 9*a^2*b*tan(d*x + c) + 2*
a^3)/tan(d*x + c)^3)/d