3.878 \(\int \frac{\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=236 \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{3 a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{17 a}{128 d (a-a \sin (c+d x))^2}+\frac{71 a}{64 d (a \sin (c+d x)+a)^2}+\frac{125}{128 d (a-a \sin (c+d x))}-\frac{5}{d (a \sin (c+d x)+a)}+\frac{5 \sin (c+d x)}{a d}+\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{1795 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

(515*Log[1 - Sin[c + d*x]])/(256*a*d) - (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + (5*Sin[c + d*x])/(a*d) - Sin[
c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (17*a)/(128*d*(a - a*Sin[c +
d*x])^2) + 125/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (3*a^2)/(16*d*(a + a*Sin[c
+ d*x])^3) + (71*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.252007, antiderivative size = 236, 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} \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{3 a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{17 a}{128 d (a-a \sin (c+d x))^2}+\frac{71 a}{64 d (a \sin (c+d x)+a)^2}+\frac{125}{128 d (a-a \sin (c+d x))}-\frac{5}{d (a \sin (c+d x)+a)}+\frac{5 \sin (c+d x)}{a d}+\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{1795 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Sin[c + d*x]^4*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(515*Log[1 - Sin[c + d*x]])/(256*a*d) - (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + (5*Sin[c + d*x])/(a*d) - Sin[
c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (17*a)/(128*d*(a - a*Sin[c +
d*x])^2) + 125/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (3*a^2)/(16*d*(a + a*Sin[c
+ d*x])^3) + (71*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(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 ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{x^{11}}{a^{11} (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^{11}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (5 a^2+\frac{a^6}{32 (a-x)^4}-\frac{17 a^5}{64 (a-x)^3}+\frac{125 a^4}{128 (a-x)^2}-\frac{515 a^3}{256 (a-x)}-a x+x^2-\frac{a^7}{16 (a+x)^5}+\frac{9 a^6}{16 (a+x)^4}-\frac{71 a^5}{32 (a+x)^3}+\frac{5 a^4}{(a+x)^2}-\frac{1795 a^3}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{1795 \log (1+\sin (c+d x))}{256 a d}+\frac{5 \sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin ^3(c+d x)}{3 a d}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{17 a}{128 d (a-a \sin (c+d x))^2}+\frac{125}{128 d (a-a \sin (c+d x))}+\frac{a^3}{64 d (a+a \sin (c+d x))^4}-\frac{3 a^2}{16 d (a+a \sin (c+d x))^3}+\frac{71 a}{64 d (a+a \sin (c+d x))^2}-\frac{5}{d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.12358, size = 153, normalized size = 0.65 \[ \frac{256 \sin ^3(c+d x)-384 \sin ^2(c+d x)+3840 \sin (c+d x)+\frac{750}{1-\sin (c+d x)}-\frac{3840}{\sin (c+d x)+1}-\frac{102}{(1-\sin (c+d x))^2}+\frac{852}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}-\frac{144}{(\sin (c+d x)+1)^3}+\frac{12}{(\sin (c+d x)+1)^4}+1545 \log (1-\sin (c+d x))-5385 \log (\sin (c+d x)+1)}{768 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(1545*Log[1 - Sin[c + d*x]] - 5385*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 102/(1 - Sin[c + d*x])^2 +
750/(1 - Sin[c + d*x]) + 3840*Sin[c + d*x] - 384*Sin[c + d*x]^2 + 256*Sin[c + d*x]^3 + 12/(1 + Sin[c + d*x])^
4 - 144/(1 + Sin[c + d*x])^3 + 852/(1 + Sin[c + d*x])^2 - 3840/(1 + Sin[c + d*x]))/(768*a*d)

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Maple [A]  time = 0.113, size = 208, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,da}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,da}}+5\,{\frac{\sin \left ( dx+c \right ) }{da}}-{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{17}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{125}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{515\,\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{3}{16\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{71}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-5\,{\frac{1}{da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{1795\,\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)^11/(a+a*sin(d*x+c)),x)

[Out]

1/3*sin(d*x+c)^3/d/a-1/2*sin(d*x+c)^2/d/a+5*sin(d*x+c)/d/a-1/96/d/a/(sin(d*x+c)-1)^3-17/128/d/a/(sin(d*x+c)-1)
^2-125/128/a/d/(sin(d*x+c)-1)+515/256/a/d*ln(sin(d*x+c)-1)+1/64/d/a/(1+sin(d*x+c))^4-3/16/d/a/(1+sin(d*x+c))^3
+71/64/a/d/(1+sin(d*x+c))^2-5/a/d/(1+sin(d*x+c))-1795/256*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.02129, size = 282, normalized size = 1.19 \begin{align*} -\frac{\frac{2 \,{\left (2295 \, \sin \left (d x + c\right )^{6} + 375 \, \sin \left (d x + c\right )^{5} - 5480 \, \sin \left (d x + c\right )^{4} - 680 \, \sin \left (d x + c\right )^{3} + 4473 \, \sin \left (d x + c\right )^{2} + 313 \, \sin \left (d x + c\right ) - 1232\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{128 \,{\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right )\right )}}{a} + \frac{5385 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{1545 \, \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(sec(d*x+c)^7*sin(d*x+c)^11/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(2295*sin(d*x + c)^6 + 375*sin(d*x + c)^5 - 5480*sin(d*x + c)^4 - 680*sin(d*x + c)^3 + 4473*sin(d*x
+ c)^2 + 313*sin(d*x + c) - 1232)/(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) - 128*(2*sin(d*x + c)^3 - 3*sin(d*x + c)^2
+ 30*sin(d*x + c))/a + 5385*log(sin(d*x + c) + 1)/a - 1545*log(sin(d*x + c) - 1)/a)/d

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Fricas [A]  time = 1.8645, size = 585, normalized size = 2.48 \begin{align*} \frac{256 \, \cos \left (d x + c\right )^{10} - 3968 \, \cos \left (d x + c\right )^{8} - 686 \, \cos \left (d x + c\right )^{6} + 2810 \, \cos \left (d x + c\right )^{4} - 796 \, \cos \left (d x + c\right )^{2} - 5385 \,{\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 ) + 1545 \,{\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 (64 \, \cos \left (d x + c\right )^{8} + 1952 \, \cos \left (d x + c\right )^{6} + 375 \, \cos \left (d x + c\right )^{4} - 70 \, \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)^11/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(256*cos(d*x + c)^10 - 3968*cos(d*x + c)^8 - 686*cos(d*x + c)^6 + 2810*cos(d*x + c)^4 - 796*cos(d*x + c)
^2 - 5385*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 1545*(cos(d*x + c)^6*sin(d*x
+ c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) + 2*(64*cos(d*x + c)^8 + 1952*cos(d*x + c)^6 + 375*cos(d*x + c)^
4 - 70*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(sec(d*x+c)**7*sin(d*x+c)**11/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.36217, size = 242, normalized size = 1.03 \begin{align*} -\frac{\frac{21540 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{6180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac{512 \,{\left (2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} + 30 \, a^{2} \sin \left (d x + c\right )\right )}}{a^{3}} + \frac{2 \,{\left (5665 \, \sin \left (d x + c\right )^{3} - 15495 \, \sin \left (d x + c\right )^{2} + 14199 \, \sin \left (d x + c\right ) - 4353\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{44875 \, \sin \left (d x + c\right )^{4} + 164140 \, \sin \left (d x + c\right )^{3} + 226578 \, \sin \left (d x + c\right )^{2} + 139660 \, \sin \left (d x + c\right ) + 32395}{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)^11/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/3072*(21540*log(abs(sin(d*x + c) + 1))/a - 6180*log(abs(sin(d*x + c) - 1))/a - 512*(2*a^2*sin(d*x + c)^3 -
3*a^2*sin(d*x + c)^2 + 30*a^2*sin(d*x + c))/a^3 + 2*(5665*sin(d*x + c)^3 - 15495*sin(d*x + c)^2 + 14199*sin(d*
x + c) - 4353)/(a*(sin(d*x + c) - 1)^3) - (44875*sin(d*x + c)^4 + 164140*sin(d*x + c)^3 + 226578*sin(d*x + c)^
2 + 139660*sin(d*x + c) + 32395)/(a*(sin(d*x + c) + 1)^4))/d