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]

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

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Rubi [A]  time = 0.20, antiderivative size = 202, normalized size of antiderivative = 1.00, 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 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]))

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]

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*}

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Mathematica [A]  time = 6.14, size = 189, normalized size = 0.94 \[ \frac {a^8 \left (-\frac {93 \log (1-\sin (c+d x))}{256 a^9}+\frac {\log (\sin (c+d x))}{a^9}-\frac {163 \log (\sin (c+d x)+1)}{256 a^9}+\frac {29}{128 a^8 (a-a \sin (c+d x))}+\frac {1}{2 a^8 (a \sin (c+d x)+a)}+\frac {7}{128 a^7 (a-a \sin (c+d x))^2}+\frac {11}{64 a^7 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^6 (a-a \sin (c+d x))^3}+\frac {1}{16 a^6 (a \sin (c+d x)+a)^3}+\frac {1}{64 a^5 (a \sin (c+d x)+a)^4}\right )}{d} \]

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

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fricas [A]  time = 0.52, size = 202, normalized size = 1.00 \[ \frac {210 \, \cos \left (d x + c\right )^{6} + 314 \, \cos \left (d x + c\right )^{4} + 164 \, \cos \left (d x + c\right )^{2} + 768 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 489 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 279 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (87 \, \cos \left (d x + c\right )^{4} + 26 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]

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)

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giac [A]  time = 0.27, size = 149, normalized size = 0.74 \[ -\frac {\frac {1956 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {1116 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {3072 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {2 \, {\left (1023 \, \sin \left (d x + c\right )^{3} - 3417 \, \sin \left (d x + c\right )^{2} + 3849 \, \sin \left (d x + c\right ) - 1471\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {4075 \, \sin \left (d x + c\right )^{4} + 17836 \, \sin \left (d x + c\right )^{3} + 29586 \, \sin \left (d x + c\right )^{2} + 22156 \, \sin \left (d x + c\right ) + 6379}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

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

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maple [A]  time = 0.45, size = 176, normalized size = 0.87 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {7}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {29}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {93 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{16 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {11}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 a d \left (1+\sin \left (d x +c \right )\right )}-\frac {163 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]

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/a/d/(sin(d*x+c)-1)^3+7/128/a/d/(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)+ln
(sin(d*x+c))/a/d+1/64/a/d/(1+sin(d*x+c))^4+1/16/a/d/(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

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maxima [A]  time = 0.33, size = 187, normalized size = 0.93 \[ \frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 87 \, \sin \left (d x + c\right )^{5} - 472 \, \sin \left (d x + c\right )^{4} + 200 \, \sin \left (d x + c\right )^{3} + 711 \, \sin \left (d x + c\right )^{2} - 121 \, \sin \left (d x + c\right ) - 400\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {489 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {279 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {768 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]

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

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mupad [B]  time = 0.16, size = 191, normalized size = 0.95 \[ \frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {163\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {93\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {-\frac {35\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {29\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {59\,{\sin \left (c+d\,x\right )}^4}{48}-\frac {25\,{\sin \left (c+d\,x\right )}^3}{48}-\frac {237\,{\sin \left (c+d\,x\right )}^2}{128}+\frac {121\,\sin \left (c+d\,x\right )}{384}+\frac {25}{24}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3-3\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^7*sin(c + d*x)*(a + a*sin(c + d*x))),x)

[Out]

log(sin(c + d*x))/(a*d) - (163*log(sin(c + d*x) + 1))/(256*a*d) - (93*log(sin(c + d*x) - 1))/(256*a*d) + ((121
*sin(c + d*x))/384 - (237*sin(c + d*x)^2)/128 - (25*sin(c + d*x)^3)/48 + (59*sin(c + d*x)^4)/48 + (29*sin(c +
d*x)^5)/128 - (35*sin(c + d*x)^6)/128 + 25/24)/(d*(a + a*sin(c + d*x) - 3*a*sin(c + d*x)^2 - 3*a*sin(c + d*x)^
3 + 3*a*sin(c + d*x)^4 + 3*a*sin(c + d*x)^5 - a*sin(c + d*x)^6 - a*sin(c + d*x)^7))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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