3.880 \(\int \frac{\sin ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=199 \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{7 a^2}{48 d (a \sin (c+d x)+a)^3}-\frac{13 a}{128 d (a-a \sin (c+d x))^2}+\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}-\frac{2}{d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}+\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{443 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

(187*Log[1 - Sin[c + d*x]])/(256*a*d) - (443*Log[1 + Sin[c + d*x]])/(256*a*d) + Sin[c + d*x]/(a*d) + a^2/(96*d
*(a - a*Sin[c + d*x])^3) - (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d
*(a + a*Sin[c + d*x])^4) - (7*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 2/(d
*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.212357, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{7 a^2}{48 d (a \sin (c+d x)+a)^3}-\frac{13 a}{128 d (a-a \sin (c+d x))^2}+\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}-\frac{2}{d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}+\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{443 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Sin[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(187*Log[1 - Sin[c + d*x]])/(256*a*d) - (443*Log[1 + Sin[c + d*x]])/(256*a*d) + Sin[c + d*x]/(a*d) + a^2/(96*d
*(a - a*Sin[c + d*x])^3) - (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d
*(a + a*Sin[c + d*x])^4) - (7*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 2/(d
*(a + a*Sin[c + d*x]))

Rule 2836

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]))

Rubi steps

\begin{align*} \int \frac{\sin ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{x^9}{a^9 (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^9}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^4}{32 (a-x)^4}-\frac{13 a^3}{64 (a-x)^3}+\frac{69 a^2}{128 (a-x)^2}-\frac{187 a}{256 (a-x)}-\frac{a^5}{16 (a+x)^5}+\frac{7 a^4}{16 (a+x)^4}-\frac{41 a^3}{32 (a+x)^3}+\frac{2 a^2}{(a+x)^2}-\frac{443 a}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{443 \log (1+\sin (c+d x))}{256 a d}+\frac{\sin (c+d x)}{a d}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{13 a}{128 d (a-a \sin (c+d x))^2}+\frac{69}{128 d (a-a \sin (c+d x))}+\frac{a^3}{64 d (a+a \sin (c+d x))^4}-\frac{7 a^2}{48 d (a+a \sin (c+d x))^3}+\frac{41 a}{64 d (a+a \sin (c+d x))^2}-\frac{2}{d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.13077, size = 133, normalized size = 0.67 \[ \frac{768 \sin (c+d x)+\frac{414}{1-\sin (c+d x)}-\frac{1536}{\sin (c+d x)+1}-\frac{78}{(1-\sin (c+d x))^2}+\frac{492}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}-\frac{112}{(\sin (c+d x)+1)^3}+\frac{12}{(\sin (c+d x)+1)^4}+561 \log (1-\sin (c+d x))-1329 \log (\sin (c+d x)+1)}{768 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(561*Log[1 - Sin[c + d*x]] - 1329*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 78/(1 - Sin[c + d*x])^2 + 4
14/(1 - Sin[c + d*x]) + 768*Sin[c + d*x] + 12/(1 + Sin[c + d*x])^4 - 112/(1 + Sin[c + d*x])^3 + 492/(1 + Sin[c
+ d*x])^2 - 1536/(1 + Sin[c + d*x]))/(768*a*d)

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Maple [A]  time = 0.105, size = 175, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{da}}-{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{13}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{69}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{187\,\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{7}{48\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{41}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{1}{da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{443\,\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(sec(d*x+c)^7*sin(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

sin(d*x+c)/d/a-1/96/d/a/(sin(d*x+c)-1)^3-13/128/d/a/(sin(d*x+c)-1)^2-69/128/a/d/(sin(d*x+c)-1)+187/256/a/d*ln(
sin(d*x+c)-1)+1/64/d/a/(1+sin(d*x+c))^4-7/48/d/a/(1+sin(d*x+c))^3+41/64/a/d/(1+sin(d*x+c))^2-2/a/d/(1+sin(d*x+
c))-443/256*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.05821, size = 251, normalized size = 1.26 \begin{align*} -\frac{\frac{2 \,{\left (975 \, \sin \left (d x + c\right )^{6} + 207 \, \sin \left (d x + c\right )^{5} - 2088 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 1569 \, \sin \left (d x + c\right )^{2} + 161 \, \sin \left (d x + c\right ) - 400\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{1329 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{561 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac{768 \, \sin \left (d x + c\right )}{a}}{768 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(975*sin(d*x + c)^6 + 207*sin(d*x + c)^5 - 2088*sin(d*x + c)^4 - 360*sin(d*x + c)^3 + 1569*sin(d*x +
c)^2 + 161*sin(d*x + c) - 400)/(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) + 1329*log(sin(d*x + c) + 1)/a - 561*log(sin(
d*x + c) - 1)/a - 768*sin(d*x + c)/a)/d

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Fricas [A]  time = 1.64101, size = 527, normalized size = 2.65 \begin{align*} -\frac{768 \, \cos \left (d x + c\right )^{8} + 1182 \, \cos \left (d x + c\right )^{6} - 1674 \, \cos \left (d x + c\right )^{4} + 636 \, \cos \left (d x + c\right )^{2} + 1329 \,{\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 ) - 561 \,{\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 (384 \, \cos \left (d x + c\right )^{6} + 207 \, \cos \left (d x + c\right )^{4} - 54 \, \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(sec(d*x+c)^7*sin(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/768*(768*cos(d*x + c)^8 + 1182*cos(d*x + c)^6 - 1674*cos(d*x + c)^4 + 636*cos(d*x + c)^2 + 1329*(cos(d*x +
c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) - 561*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)
*log(-sin(d*x + c) + 1) - 2*(384*cos(d*x + c)^6 + 207*cos(d*x + c)^4 - 54*cos(d*x + c)^2 + 8)*sin(d*x + c) - 1
12)/(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(sec(d*x+c)**7*sin(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.34441, size = 198, normalized size = 0.99 \begin{align*} -\frac{\frac{5316 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{2244 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac{3072 \, \sin \left (d x + c\right )}{a} + \frac{2 \,{\left (2057 \, \sin \left (d x + c\right )^{3} - 5343 \, \sin \left (d x + c\right )^{2} + 4671 \, \sin \left (d x + c\right ) - 1369\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{11075 \, \sin \left (d x + c\right )^{4} + 38156 \, \sin \left (d x + c\right )^{3} + 49986 \, \sin \left (d x + c\right )^{2} + 29356 \, \sin \left (d x + c\right ) + 6499}{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(sec(d*x+c)^7*sin(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/3072*(5316*log(abs(sin(d*x + c) + 1))/a - 2244*log(abs(sin(d*x + c) - 1))/a - 3072*sin(d*x + c)/a + 2*(2057
*sin(d*x + c)^3 - 5343*sin(d*x + c)^2 + 4671*sin(d*x + c) - 1369)/(a*(sin(d*x + c) - 1)^3) - (11075*sin(d*x +
c)^4 + 38156*sin(d*x + c)^3 + 49986*sin(d*x + c)^2 + 29356*sin(d*x + c) + 6499)/(a*(sin(d*x + c) + 1)^4))/d