3.82 \(\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=195 \[ \frac{a^2}{48 d (a \sin (c+d x)+a)^6}-\frac{3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac{1}{256 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac{5}{256 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}-\frac{7 a}{80 d (a \sin (c+d x)+a)^5}+\frac{1}{8 d (a \sin (c+d x)+a)^4}-\frac{5}{96 a d (a \sin (c+d x)+a)^3} \]

[Out]

-ArcTanh[Sin[c + d*x]]/(128*a^4*d) + a^2/(48*d*(a + a*Sin[c + d*x])^6) - (7*a)/(80*d*(a + a*Sin[c + d*x])^5) +
1/(8*d*(a + a*Sin[c + d*x])^4) - 5/(96*a*d*(a + a*Sin[c + d*x])^3) + 1/(256*d*(a^2 - a^2*Sin[c + d*x])^2) - 5
/(256*d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(256*d*(a^4 - a^4*Sin[c + d*x])) - 1/(256*d*(a^4 + a^4*Sin[c + d*x]))

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Rubi [A]  time = 0.134569, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2707, 88, 206} \[ \frac{a^2}{48 d (a \sin (c+d x)+a)^6}-\frac{3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac{1}{256 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac{5}{256 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}-\frac{7 a}{80 d (a \sin (c+d x)+a)^5}+\frac{1}{8 d (a \sin (c+d x)+a)^4}-\frac{5}{96 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

-ArcTanh[Sin[c + d*x]]/(128*a^4*d) + a^2/(48*d*(a + a*Sin[c + d*x])^6) - (7*a)/(80*d*(a + a*Sin[c + d*x])^5) +
1/(8*d*(a + a*Sin[c + d*x])^4) - 5/(96*a*d*(a + a*Sin[c + d*x])^3) + 1/(256*d*(a^2 - a^2*Sin[c + d*x])^2) - 5
/(256*d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(256*d*(a^4 - a^4*Sin[c + d*x])) - 1/(256*d*(a^4 + a^4*Sin[c + d*x]))

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a-x)^3 (a+x)^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{128 a^2 (a-x)^3}-\frac{3}{256 a^3 (a-x)^2}-\frac{a^2}{8 (a+x)^7}+\frac{7 a}{16 (a+x)^6}-\frac{1}{2 (a+x)^5}+\frac{5}{32 a (a+x)^4}+\frac{5}{128 a^2 (a+x)^3}+\frac{1}{256 a^3 (a+x)^2}-\frac{1}{128 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2}{48 d (a+a \sin (c+d x))^6}-\frac{7 a}{80 d (a+a \sin (c+d x))^5}+\frac{1}{8 d (a+a \sin (c+d x))^4}-\frac{5}{96 a d (a+a \sin (c+d x))^3}+\frac{1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac{5}{256 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac{1}{256 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a^3 d}\\ &=-\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}+\frac{a^2}{48 d (a+a \sin (c+d x))^6}-\frac{7 a}{80 d (a+a \sin (c+d x))^5}+\frac{1}{8 d (a+a \sin (c+d x))^4}-\frac{5}{96 a d (a+a \sin (c+d x))^3}+\frac{1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac{5}{256 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac{1}{256 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.48884, size = 112, normalized size = 0.57 \[ -\frac{30 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left (15 \sin ^7(c+d x)+60 \sin ^6(c+d x)+65 \sin ^5(c+d x)+440 \sin ^4(c+d x)+257 \sin ^3(c+d x)-132 \sin ^2(c+d x)-177 \sin (c+d x)-48\right )}{(\sin (c+d x)-1)^2 (\sin (c+d x)+1)^6}}{3840 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

-(30*ArcTanh[Sin[c + d*x]] - (2*(-48 - 177*Sin[c + d*x] - 132*Sin[c + d*x]^2 + 257*Sin[c + d*x]^3 + 440*Sin[c
+ d*x]^4 + 65*Sin[c + d*x]^5 + 60*Sin[c + d*x]^6 + 15*Sin[c + d*x]^7))/((-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x
])^6))/(3840*a^4*d)

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Maple [A]  time = 0.088, size = 180, normalized size = 0.9 \begin{align*}{\frac{1}{256\,d{a}^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}+{\frac{3}{256\,d{a}^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{256\,d{a}^{4}}}+{\frac{1}{48\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{7}{80\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{8\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5}{96\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5}{256\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{256\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5/(a+a*sin(d*x+c))^4,x)

[Out]

1/256/d/a^4/(sin(d*x+c)-1)^2+3/256/d/a^4/(sin(d*x+c)-1)+1/256/d/a^4*ln(sin(d*x+c)-1)+1/48/d/a^4/(1+sin(d*x+c))
^6-7/80/d/a^4/(1+sin(d*x+c))^5+1/8/d/a^4/(1+sin(d*x+c))^4-5/96/d/a^4/(1+sin(d*x+c))^3-5/256/d/a^4/(1+sin(d*x+c
))^2-1/256/d/a^4/(1+sin(d*x+c))-1/256*ln(1+sin(d*x+c))/a^4/d

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Maxima [A]  time = 3.47638, size = 288, normalized size = 1.48 \begin{align*} \frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{7} + 60 \, \sin \left (d x + c\right )^{6} + 65 \, \sin \left (d x + c\right )^{5} + 440 \, \sin \left (d x + c\right )^{4} + 257 \, \sin \left (d x + c\right )^{3} - 132 \, \sin \left (d x + c\right )^{2} - 177 \, \sin \left (d x + c\right ) - 48\right )}}{a^{4} \sin \left (d x + c\right )^{8} + 4 \, a^{4} \sin \left (d x + c\right )^{7} + 4 \, a^{4} \sin \left (d x + c\right )^{6} - 4 \, a^{4} \sin \left (d x + c\right )^{5} - 10 \, a^{4} \sin \left (d x + c\right )^{4} - 4 \, a^{4} \sin \left (d x + c\right )^{3} + 4 \, a^{4} \sin \left (d x + c\right )^{2} + 4 \, a^{4} \sin \left (d x + c\right ) + a^{4}} - \frac{15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4}}}{3840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3840*(2*(15*sin(d*x + c)^7 + 60*sin(d*x + c)^6 + 65*sin(d*x + c)^5 + 440*sin(d*x + c)^4 + 257*sin(d*x + c)^3
- 132*sin(d*x + c)^2 - 177*sin(d*x + c) - 48)/(a^4*sin(d*x + c)^8 + 4*a^4*sin(d*x + c)^7 + 4*a^4*sin(d*x + c)
^6 - 4*a^4*sin(d*x + c)^5 - 10*a^4*sin(d*x + c)^4 - 4*a^4*sin(d*x + c)^3 + 4*a^4*sin(d*x + c)^2 + 4*a^4*sin(d*
x + c) + a^4) - 15*log(sin(d*x + c) + 1)/a^4 + 15*log(sin(d*x + c) - 1)/a^4)/d

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Fricas [A]  time = 1.73448, size = 775, normalized size = 3.97 \begin{align*} -\frac{120 \, \cos \left (d x + c\right )^{6} - 1240 \, \cos \left (d x + c\right )^{4} + 1856 \, \cos \left (d x + c\right )^{2} + 15 \,{\left (\cos \left (d x + c\right )^{8} - 8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{4} - 4 \,{\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (\cos \left (d x + c\right )^{8} - 8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{4} - 4 \,{\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (15 \, \cos \left (d x + c\right )^{6} - 110 \, \cos \left (d x + c\right )^{4} + 432 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 640}{3840 \,{\left (a^{4} d \cos \left (d x + c\right )^{8} - 8 \, a^{4} d \cos \left (d x + c\right )^{6} + 8 \, a^{4} d \cos \left (d x + c\right )^{4} - 4 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 2 \, a^{4} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3840*(120*cos(d*x + c)^6 - 1240*cos(d*x + c)^4 + 1856*cos(d*x + c)^2 + 15*(cos(d*x + c)^8 - 8*cos(d*x + c)^
6 + 8*cos(d*x + c)^4 - 4*(cos(d*x + c)^6 - 2*cos(d*x + c)^4)*sin(d*x + c))*log(sin(d*x + c) + 1) - 15*(cos(d*x
+ c)^8 - 8*cos(d*x + c)^6 + 8*cos(d*x + c)^4 - 4*(cos(d*x + c)^6 - 2*cos(d*x + c)^4)*sin(d*x + c))*log(-sin(d
*x + c) + 1) + 2*(15*cos(d*x + c)^6 - 110*cos(d*x + c)^4 + 432*cos(d*x + c)^2 - 160)*sin(d*x + c) - 640)/(a^4*
d*cos(d*x + c)^8 - 8*a^4*d*cos(d*x + c)^6 + 8*a^4*d*cos(d*x + c)^4 - 4*(a^4*d*cos(d*x + c)^6 - 2*a^4*d*cos(d*x
+ c)^4)*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(tan(d*x+c)**5/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 5.53021, size = 197, normalized size = 1.01 \begin{align*} -\frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4}} + \frac{30 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 7\right )}}{a^{4}{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac{147 \, \sin \left (d x + c\right )^{6} + 822 \, \sin \left (d x + c\right )^{5} + 1605 \, \sin \left (d x + c\right )^{4} + 340 \, \sin \left (d x + c\right )^{3} - 675 \, \sin \left (d x + c\right )^{2} - 522 \, \sin \left (d x + c\right ) - 117}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{6}}}{15360 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/15360*(60*log(abs(sin(d*x + c) + 1))/a^4 - 60*log(abs(sin(d*x + c) - 1))/a^4 + 30*(3*sin(d*x + c)^2 - 12*si
n(d*x + c) + 7)/(a^4*(sin(d*x + c) - 1)^2) - (147*sin(d*x + c)^6 + 822*sin(d*x + c)^5 + 1605*sin(d*x + c)^4 +
340*sin(d*x + c)^3 - 675*sin(d*x + c)^2 - 522*sin(d*x + c) - 117)/(a^4*(sin(d*x + c) + 1)^6))/d