3.53 \(\int \frac{\tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=84 \[ \frac{\tan ^7(c+d x)}{7 a d}-\frac{\sec ^7(c+d x)}{7 a d}+\frac{3 \sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{a d}+\frac{\sec (c+d x)}{a d} \]

[Out]

Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^7/(7*a*d) + Tan[c + d*x]
^7/(7*a*d)

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Rubi [A]  time = 0.0973616, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2606, 194} \[ \frac{\tan ^7(c+d x)}{7 a d}-\frac{\sec ^7(c+d x)}{7 a d}+\frac{3 \sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{a d}+\frac{\sec (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^6/(a + a*Sin[c + d*x]),x]

[Out]

Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^7/(7*a*d) + Tan[c + d*x]
^7/(7*a*d)

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a}-\frac{\int \sec (c+d x) \tan ^7(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{\tan ^7(c+d x)}{7 a d}-\frac{\operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{\sec (c+d x)}{a d}-\frac{\sec ^3(c+d x)}{a d}+\frac{3 \sec ^5(c+d x)}{5 a d}-\frac{\sec ^7(c+d x)}{7 a d}+\frac{\tan ^7(c+d x)}{7 a d}\\ \end{align*}

Mathematica [A]  time = 0.319019, size = 146, normalized size = 1.74 \[ \frac{\sec ^5(c+d x) (2432 \sin (c+d x)-1905 \sin (2 (c+d x))+320 \sin (3 (c+d x))-1524 \sin (4 (c+d x))+960 \sin (5 (c+d x))-381 \sin (6 (c+d x))-7620 \cos (c+d x)+3760 \cos (2 (c+d x))-3810 \cos (3 (c+d x))+1440 \cos (4 (c+d x))-762 \cos (5 (c+d x))+80 \cos (6 (c+d x))+2912)}{17920 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^6/(a + a*Sin[c + d*x]),x]

[Out]

(Sec[c + d*x]^5*(2912 - 7620*Cos[c + d*x] + 3760*Cos[2*(c + d*x)] - 3810*Cos[3*(c + d*x)] + 1440*Cos[4*(c + d*
x)] - 762*Cos[5*(c + d*x)] + 80*Cos[6*(c + d*x)] + 2432*Sin[c + d*x] - 1905*Sin[2*(c + d*x)] + 320*Sin[3*(c +
d*x)] - 1524*Sin[4*(c + d*x)] + 960*Sin[5*(c + d*x)] - 381*Sin[6*(c + d*x)]))/(17920*a*d*(1 + Sin[c + d*x]))

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Maple [B]  time = 0.057, size = 175, normalized size = 2.1 \begin{align*} 128\,{\frac{1}{da} \left ( -{\frac{1}{1280\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{5}}}-{\frac{1}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{4}}}+{\frac{1}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}-{\frac{5}{2048\,\tan \left ( 1/2\,dx+c/2 \right ) -2048}}-{\frac{1}{448\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+{\frac{1}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{9}{1280\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}-{\frac{1}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{1}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{3}{1024\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{5}{2048\,\tan \left ( 1/2\,dx+c/2 \right ) +2048}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^6/(a+a*sin(d*x+c)),x)

[Out]

128/d/a*(-1/1280/(tan(1/2*d*x+1/2*c)-1)^5-1/512/(tan(1/2*d*x+1/2*c)-1)^4+1/512/(tan(1/2*d*x+1/2*c)-1)^2-5/2048
/(tan(1/2*d*x+1/2*c)-1)-1/448/(tan(1/2*d*x+1/2*c)+1)^7+1/128/(tan(1/2*d*x+1/2*c)+1)^6-9/1280/(tan(1/2*d*x+1/2*
c)+1)^5-1/512/(tan(1/2*d*x+1/2*c)+1)^4+1/512/(tan(1/2*d*x+1/2*c)+1)^3+3/1024/(tan(1/2*d*x+1/2*c)+1)^2+5/2048/(
tan(1/2*d*x+1/2*c)+1))

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Maxima [B]  time = 1.09477, size = 456, normalized size = 5.43 \begin{align*} \frac{32 \,{\left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 1\right )}}{35 \,{\left (a + \frac{2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{10 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{5 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{20 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{5 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{10 \, a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{4 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{2 \, a \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

32/35*(2*sin(d*x + c)/(cos(d*x + c) + 1) - 4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 10*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 + 5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1)/((a + 2*a*si
n(d*x + c)/(cos(d*x + c) + 1) - 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 10*a*sin(d*x + c)^3/(cos(d*x + c) +
1)^3 + 5*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 20*a*sin(d*x + c)^
7/(cos(d*x + c) + 1)^7 - 5*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 10*a*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 +
4*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 2*a*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - a*sin(d*x + c)^12/(cos
(d*x + c) + 1)^12)*d)

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Fricas [A]  time = 1.67288, size = 248, normalized size = 2.95 \begin{align*} \frac{5 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1}{35 \,{\left (a d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/35*(5*cos(d*x + c)^6 + 15*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 2*(15*cos(d*x + c)^4 - 10*cos(d*x + c)^2 + 3)*
sin(d*x + c) + 1)/(a*d*cos(d*x + c)^5*sin(d*x + c) + a*d*cos(d*x + c)^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{6}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**6/(a+a*sin(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**6/(sin(c + d*x) + 1), x)/a

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Giac [B]  time = 5.31986, size = 232, normalized size = 2.76 \begin{align*} -\frac{\frac{7 \,{\left (25 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 33\right )}}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{5}} - \frac{175 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1260 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3815 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6020 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4641 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1792 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 281}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/560*(7*(25*tan(1/2*d*x + 1/2*c)^4 - 120*tan(1/2*d*x + 1/2*c)^3 + 210*tan(1/2*d*x + 1/2*c)^2 - 140*tan(1/2*d
*x + 1/2*c) + 33)/(a*(tan(1/2*d*x + 1/2*c) - 1)^5) - (175*tan(1/2*d*x + 1/2*c)^6 + 1260*tan(1/2*d*x + 1/2*c)^5
+ 3815*tan(1/2*d*x + 1/2*c)^4 + 6020*tan(1/2*d*x + 1/2*c)^3 + 4641*tan(1/2*d*x + 1/2*c)^2 + 1792*tan(1/2*d*x
+ 1/2*c) + 281)/(a*(tan(1/2*d*x + 1/2*c) + 1)^7))/d