\(\int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx\) [38]

Optimal result

Integrand size = 21, antiderivative size = 113 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a \left (a^2+3 b^2\right ) \cot (c+d x)}{d}-\frac {3 a^2 b \cot ^2(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3597, 908} \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a \left (a^2+3 b^2\right ) \cot (c+d x)}{d}+\frac {b \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{d}-\frac {3 a^2 b \cot ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]

[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 908

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]))

Rule 3597

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]

Rubi steps \begin{align*} \text {integral}& = \frac {b \text {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 \text {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] (verified)

Time = 4.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.88 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(b+a \cot (c+d x))^3 \sec ^2(c+d x) \left (-16 a^3 \cos (c+d x)-2 \sin (c+d x) \left (18 a^2 b-6 b^3+6 \left (3 a^2 b+b^3\right ) \cos (2 (c+d x))+9 a^2 b \log (\cos (c+d x))+3 b^3 \log (\cos (c+d x))-3 b \left (3 a^2+b^2\right ) \cos (4 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))-9 a^2 b \log (\sin (c+d x))-3 b^3 \log (\sin (c+d x))+2 a^3 \sin (4 (c+d x))+18 a b^2 \sin (4 (c+d x))\right )\right )}{48 d (a \cos (c+d x)+b \sin (c+d x))^3} \]

[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)

Maple [A] (verified)

Time = 6.97 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{2} b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\)	\(106\)
default	\(\frac {b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{2} b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\)	\(106\)
risch	\(\frac {6 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+\frac {4 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{3}+12 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-6 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+\frac {20 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{3}-6 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+4 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-\frac {4 i a^{3}}{3}-6 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-12 i a \,b^{2}+12 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(337\)

[In]

int(csc(d*x+c)^4*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (107) = 214\).

Time = 0.28 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.10 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {4 \, {\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 18 \, a b^{2} \cos \left (d x + c\right ) - 6 \, {\left (a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) + 3 \, {\left (b^{3} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F]

\[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \csc ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(csc(d*x+c)**4*(a+b*tan(d*x+c))**3,x)

[Out]

Integral((a + b*tan(c + d*x))**3*csc(c + d*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 \, b^{3} \tan \left (d x + c\right )^{2} + 18 \, a b^{2} \tan \left (d x + c\right ) + 6 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {9 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3} + 6 \, {\left (a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 \, b^{3} \tan \left (d x + c\right )^{2} + 18 \, a b^{2} \tan \left (d x + c\right ) + 6 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac {33 \, a^{2} b \tan \left (d x + c\right )^{3} + 11 \, b^{3} \tan \left (d x + c\right )^{3} + 6 \, a^{3} \tan \left (d x + c\right )^{2} + 18 \, a b^{2} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2\,b+b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {a^3}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^3+3\,a\,b^2\right )+\frac {3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )}{2}\right )}{d}+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}+\frac {3\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d} \]

[In]

int((a + b*tan(c + d*x))^3/sin(c + d*x)^4,x)

[Out]

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