3.44 \(\int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=130 \[ \frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d} \]

[Out]

(-35*ArcTanh[Sin[c + d*x]])/(128*a*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*
x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d
*x]^8/(8*a*d)

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Rubi [A]  time = 0.161036, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2611, 3770} \[ \frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

(-35*ArcTanh[Sin[c + d*x]])/(128*a*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*
x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d
*x]^8/(8*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 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^2(c+d x) \tan ^7(c+d x) \, dx}{a}-\frac{\int \sec (c+d x) \tan ^8(c+d x) \, dx}{a}\\ &=-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{7 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^7 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{48 a}\\ &=-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}+\frac{35 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{64 a}\\ &=\frac{35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \int \sec (c+d x) \, dx}{128 a}\\ &=-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}\\ \end{align*}

Mathematica [A]  time = 1.00892, size = 101, normalized size = 0.78 \[ -\frac{\frac{279 \sin ^6(c+d x)+87 \sin ^5(c+d x)-424 \sin ^4(c+d x)-136 \sin ^3(c+d x)+249 \sin ^2(c+d x)+57 \sin (c+d x)-48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+105 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

-(105*ArcTanh[Sin[c + d*x]] + (-48 + 57*Sin[c + d*x] + 249*Sin[c + d*x]^2 - 136*Sin[c + d*x]^3 - 424*Sin[c + d
*x]^4 + 87*Sin[c + d*x]^5 + 279*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(384*a*d)

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Maple [A]  time = 0.059, size = 162, normalized size = 1.3 \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{9}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{29}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{35\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{256\,da}}+{\frac{1}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5}{48\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{19}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{35\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^7/(a+a*sin(d*x+c)),x)

[Out]

-1/96/d/a/(sin(d*x+c)-1)^3-9/128/d/a/(sin(d*x+c)-1)^2-29/128/a/d/(sin(d*x+c)-1)+35/256/a/d*ln(sin(d*x+c)-1)+1/
64/d/a/(1+sin(d*x+c))^4-5/48/d/a/(1+sin(d*x+c))^3+19/64/a/d/(1+sin(d*x+c))^2-1/2/a/d/(1+sin(d*x+c))-35/256*ln(
1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.02832, size = 236, normalized size = 1.82 \begin{align*} -\frac{\frac{2 \,{\left (279 \, \sin \left (d x + c\right )^{6} + 87 \, \sin \left (d x + c\right )^{5} - 424 \, \sin \left (d x + c\right )^{4} - 136 \, \sin \left (d x + c\right )^{3} + 249 \, \sin \left (d x + c\right )^{2} + 57 \, \sin \left (d x + c\right ) - 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} + \frac{105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(279*sin(d*x + c)^6 + 87*sin(d*x + c)^5 - 424*sin(d*x + c)^4 - 136*sin(d*x + c)^3 + 249*sin(d*x + c)
^2 + 57*sin(d*x + c) - 48)/(a*sin(d*x + c)^7 + a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 - 3*a*sin(d*x + c)^4 + 3*
a*sin(d*x + c)^3 + 3*a*sin(d*x + c)^2 - a*sin(d*x + c) - a) + 105*log(sin(d*x + c) + 1)/a - 105*log(sin(d*x +
c) - 1)/a)/d

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Fricas [A]  time = 2.05396, size = 464, normalized size = 3.57 \begin{align*} -\frac{558 \, \cos \left (d x + c\right )^{6} - 826 \, \cos \left (d x + c\right )^{4} + 476 \, \cos \left (d x + c\right )^{2} + 105 \,{\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (87 \, \cos \left (d x + c\right )^{4} - 38 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \,{\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/768*(558*cos(d*x + c)^6 - 826*cos(d*x + c)^4 + 476*cos(d*x + c)^2 + 105*(cos(d*x + c)^6*sin(d*x + c) + cos(
d*x + c)^6)*log(sin(d*x + c) + 1) - 105*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1)
- 2*(87*cos(d*x + c)^4 - 38*cos(d*x + c)^2 + 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos
(d*x + c)^6)

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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)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 8.06961, size = 184, normalized size = 1.42 \begin{align*} -\frac{\frac{420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (385 \, \sin \left (d x + c\right )^{3} - 807 \, \sin \left (d x + c\right )^{2} + 567 \, \sin \left (d x + c\right ) - 129\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{875 \, \sin \left (d x + c\right )^{4} + 1964 \, \sin \left (d x + c\right )^{3} + 1554 \, \sin \left (d x + c\right )^{2} + 396 \, \sin \left (d x + c\right ) - 21}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/3072*(420*log(abs(sin(d*x + c) + 1))/a - 420*log(abs(sin(d*x + c) - 1))/a + 2*(385*sin(d*x + c)^3 - 807*sin
(d*x + c)^2 + 567*sin(d*x + c) - 129)/(a*(sin(d*x + c) - 1)^3) - (875*sin(d*x + c)^4 + 1964*sin(d*x + c)^3 + 1
554*sin(d*x + c)^2 + 396*sin(d*x + c) - 21)/(a*(sin(d*x + c) + 1)^4))/d