3.890 \(\int \frac{\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=202 \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{7 a}{128 d (a-a \sin (c+d x))^2}+\frac{11 a}{64 d (a \sin (c+d x)+a)^2}+\frac{29}{128 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{163 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

(-93*Log[1 - Sin[c + d*x]])/(256*a*d) + Log[Sin[c + d*x]]/(a*d) - (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/
(96*d*(a - a*Sin[c + d*x])^3) + (7*a)/(128*d*(a - a*Sin[c + d*x])^2) + 29/(128*d*(a - a*Sin[c + d*x])) + a^3/(
64*d*(a + a*Sin[c + d*x])^4) + a^2/(16*d*(a + a*Sin[c + d*x])^3) + (11*a)/(64*d*(a + a*Sin[c + d*x])^2) + 1/(2
*d*(a + a*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.200673, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{7 a}{128 d (a-a \sin (c+d x))^2}+\frac{11 a}{64 d (a \sin (c+d x)+a)^2}+\frac{29}{128 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{163 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-93*Log[1 - Sin[c + d*x]])/(256*a*d) + Log[Sin[c + d*x]]/(a*d) - (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/
(96*d*(a - a*Sin[c + d*x])^3) + (7*a)/(128*d*(a - a*Sin[c + d*x])^2) + 29/(128*d*(a - a*Sin[c + d*x])) + a^3/(
64*d*(a + a*Sin[c + d*x])^4) + a^2/(16*d*(a + a*Sin[c + d*x])^3) + (11*a)/(64*d*(a + a*Sin[c + d*x])^2) + 1/(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{\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{a}{(a-x)^4 x (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \frac{1}{(a-x)^4 x (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{1}{32 a^6 (a-x)^4}+\frac{7}{64 a^7 (a-x)^3}+\frac{29}{128 a^8 (a-x)^2}+\frac{93}{256 a^9 (a-x)}+\frac{1}{a^9 x}-\frac{1}{16 a^5 (a+x)^5}-\frac{3}{16 a^6 (a+x)^4}-\frac{11}{32 a^7 (a+x)^3}-\frac{1}{2 a^8 (a+x)^2}-\frac{163}{256 a^9 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{163 \log (1+\sin (c+d x))}{256 a d}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{7 a}{128 d (a-a \sin (c+d x))^2}+\frac{29}{128 d (a-a \sin (c+d x))}+\frac{a^3}{64 d (a+a \sin (c+d x))^4}+\frac{a^2}{16 d (a+a \sin (c+d x))^3}+\frac{11 a}{64 d (a+a \sin (c+d x))^2}+\frac{1}{2 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.12995, size = 189, normalized size = 0.94 \[ \frac{a^8 \left (\frac{29}{128 a^8 (a-a \sin (c+d x))}+\frac{1}{2 a^8 (a \sin (c+d x)+a)}+\frac{7}{128 a^7 (a-a \sin (c+d x))^2}+\frac{11}{64 a^7 (a \sin (c+d x)+a)^2}+\frac{1}{96 a^6 (a-a \sin (c+d x))^3}+\frac{1}{16 a^6 (a \sin (c+d x)+a)^3}+\frac{1}{64 a^5 (a \sin (c+d x)+a)^4}-\frac{93 \log (1-\sin (c+d x))}{256 a^9}+\frac{\log (\sin (c+d x))}{a^9}-\frac{163 \log (\sin (c+d x)+1)}{256 a^9}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*((-93*Log[1 - Sin[c + d*x]])/(256*a^9) + Log[Sin[c + d*x]]/a^9 - (163*Log[1 + Sin[c + d*x]])/(256*a^9) +
1/(96*a^6*(a - a*Sin[c + d*x])^3) + 7/(128*a^7*(a - a*Sin[c + d*x])^2) + 29/(128*a^8*(a - a*Sin[c + d*x])) + 1
/(64*a^5*(a + a*Sin[c + d*x])^4) + 1/(16*a^6*(a + a*Sin[c + d*x])^3) + 11/(64*a^7*(a + a*Sin[c + d*x])^2) + 1/
(2*a^8*(a + a*Sin[c + d*x]))))/d

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 176, normalized size = 0.9 \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{7}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{29}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{93\,\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{1}{16\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{11}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{163\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96/d/a/(sin(d*x+c)-1)^3+7/128/d/a/(sin(d*x+c)-1)^2-29/128/a/d/(sin(d*x+c)-1)-93/256/a/d*ln(sin(d*x+c)-1)+1/
64/d/a/(1+sin(d*x+c))^4+1/16/d/a/(1+sin(d*x+c))^3+11/64/a/d/(1+sin(d*x+c))^2+1/2/a/d/(1+sin(d*x+c))-163/256*ln
(1+sin(d*x+c))/a/d+ln(sin(d*x+c))/a/d

________________________________________________________________________________________

Maxima [A]  time = 1.02196, size = 252, normalized size = 1.25 \begin{align*} \frac{\frac{2 \,{\left (105 \, \sin \left (d x + c\right )^{6} - 87 \, \sin \left (d x + c\right )^{5} - 472 \, \sin \left (d x + c\right )^{4} + 200 \, \sin \left (d x + c\right )^{3} + 711 \, \sin \left (d x + c\right )^{2} - 121 \, \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{489 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{279 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac{768 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(2*(105*sin(d*x + c)^6 - 87*sin(d*x + c)^5 - 472*sin(d*x + c)^4 + 200*sin(d*x + c)^3 + 711*sin(d*x + c)^
2 - 121*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) - 489*log(sin(d*x + c) + 1)/a - 279*log(sin(d*x +
c) - 1)/a + 768*log(sin(d*x + c))/a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.64071, size = 564, normalized size = 2.79 \begin{align*} \frac{210 \, \cos \left (d x + c\right )^{6} + 314 \, \cos \left (d x + c\right )^{4} + 164 \, \cos \left (d x + c\right )^{2} + 768 \,{\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 489 \,{\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 ) - 279 \,{\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} + 26 \, \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(csc(d*x+c)*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(210*cos(d*x + c)^6 + 314*cos(d*x + c)^4 + 164*cos(d*x + c)^2 + 768*(cos(d*x + c)^6*sin(d*x + c) + cos(d
*x + c)^6)*log(1/2*sin(d*x + c)) - 489*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) -
279*(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 + 26*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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.34404, size = 201, normalized size = 1. \begin{align*} -\frac{\frac{1956 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac{1116 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac{3072 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{2 \,{\left (1023 \, \sin \left (d x + c\right )^{3} - 3417 \, \sin \left (d x + c\right )^{2} + 3849 \, \sin \left (d x + c\right ) - 1471\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{4075 \, \sin \left (d x + c\right )^{4} + 17836 \, \sin \left (d x + c\right )^{3} + 29586 \, \sin \left (d x + c\right )^{2} + 22156 \, \sin \left (d x + c\right ) + 6379}{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(csc(d*x+c)*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/3072*(1956*log(abs(sin(d*x + c) + 1))/a + 1116*log(abs(sin(d*x + c) - 1))/a - 3072*log(abs(sin(d*x + c)))/a
- 2*(1023*sin(d*x + c)^3 - 3417*sin(d*x + c)^2 + 3849*sin(d*x + c) - 1471)/(a*(sin(d*x + c) - 1)^3) - (4075*s
in(d*x + c)^4 + 17836*sin(d*x + c)^3 + 29586*sin(d*x + c)^2 + 22156*sin(d*x + c) + 6379)/(a*(sin(d*x + c) + 1)
^4))/d