3.892 \(\int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=232 \[ \frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {5 a^2}{48 d (a \sin (c+d x)+a)^3}+\frac {11 a}{128 d (a-a \sin (c+d x))^2}+\frac {29 a}{64 d (a \sin (c+d x)+a)^2}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {2}{d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

csc(d*x+c)/a/d-1/2*csc(d*x+c)^2/a/d-325/256*ln(1-sin(d*x+c))/a/d+5*ln(sin(d*x+c))/a/d-955/256*ln(1+sin(d*x+c))
/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3+11/128*a/d/(a-a*sin(d*x+c))^2+69/128/d/(a-a*sin(d*x+c))+1/64*a^3/d/(a+a*sin
(d*x+c))^4+5/48*a^2/d/(a+a*sin(d*x+c))^3+29/64*a/d/(a+a*sin(d*x+c))^2+2/d/(a+a*sin(d*x+c))

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Rubi [A]  time = 0.25, antiderivative size = 232, normalized size of antiderivative = 1.00, 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} \[ \frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {5 a^2}{48 d (a \sin (c+d x)+a)^3}+\frac {11 a}{128 d (a-a \sin (c+d x))^2}+\frac {29 a}{64 d (a \sin (c+d x)+a)^2}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {2}{d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (325*Log[1 - Sin[c + d*x]])/(256*a*d) + (5*Log[Sin[c + d*x]])/(a
*d) - (955*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (11*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) + (5*a^2)/(48*d*(a + a*Sin[c +
d*x])^3) + (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 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 ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {a^3}{(a-x)^4 x^3 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{10} \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 x^3 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{10} \operatorname {Subst}\left (\int \left (\frac {1}{32 a^8 (a-x)^4}+\frac {11}{64 a^9 (a-x)^3}+\frac {69}{128 a^{10} (a-x)^2}+\frac {325}{256 a^{11} (a-x)}+\frac {1}{a^9 x^3}-\frac {1}{a^{10} x^2}+\frac {5}{a^{11} x}-\frac {1}{16 a^7 (a+x)^5}-\frac {5}{16 a^8 (a+x)^4}-\frac {29}{32 a^9 (a+x)^3}-\frac {2}{a^{10} (a+x)^2}-\frac {955}{256 a^{11} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {11 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 {5 a^2}{48 d (a+a \sin (c+d x))^3}+\frac {29 a}{64 d (a+a \sin (c+d x))^2}+\frac {2}{d (a+a \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.19, size = 213, normalized size = 0.92 \[ \frac {a^{10} \left (-\frac {\csc ^2(c+d x)}{2 a^{11}}+\frac {\csc (c+d x)}{a^{11}}-\frac {325 \log (1-\sin (c+d x))}{256 a^{11}}+\frac {5 \log (\sin (c+d x))}{a^{11}}-\frac {955 \log (\sin (c+d x)+1)}{256 a^{11}}+\frac {69}{128 a^{10} (a-a \sin (c+d x))}+\frac {2}{a^{10} (a \sin (c+d x)+a)}+\frac {11}{128 a^9 (a-a \sin (c+d x))^2}+\frac {29}{64 a^9 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^8 (a-a \sin (c+d x))^3}+\frac {5}{48 a^8 (a \sin (c+d x)+a)^3}+\frac {1}{64 a^7 (a \sin (c+d x)+a)^4}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(a^10*(Csc[c + d*x]/a^11 - Csc[c + d*x]^2/(2*a^11) - (325*Log[1 - Sin[c + d*x]])/(256*a^11) + (5*Log[Sin[c + d
*x]])/a^11 - (955*Log[1 + Sin[c + d*x]])/(256*a^11) + 1/(96*a^8*(a - a*Sin[c + d*x])^3) + 11/(128*a^9*(a - a*S
in[c + d*x])^2) + 69/(128*a^10*(a - a*Sin[c + d*x])) + 1/(64*a^7*(a + a*Sin[c + d*x])^4) + 5/(48*a^8*(a + a*Si
n[c + d*x])^3) + 29/(64*a^9*(a + a*Sin[c + d*x])^2) + 2/(a^10*(a + a*Sin[c + d*x]))))/d

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fricas [A]  time = 0.53, size = 311, normalized size = 1.34 \[ \frac {1890 \, \cos \left (d x + c\right )^{8} - 600 \, \cos \left (d x + c\right )^{6} - 582 \, \cos \left (d x + c\right )^{4} - 212 \, \cos \left (d x + c\right )^{2} + 3840 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2865 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 975 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 165 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6} + {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(1890*cos(d*x + c)^8 - 600*cos(d*x + c)^6 - 582*cos(d*x + c)^4 - 212*cos(d*x + c)^2 + 3840*(cos(d*x + c)
^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(1/2*sin(d*x + c)) - 2865*(cos(d*x +
c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(sin(d*x + c) + 1) - 975*(cos(d*x +
c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(15*cos(d*
x + c)^6 - 165*cos(d*x + c)^4 - 34*cos(d*x + c)^2 - 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^8 - a*d*cos(d*x +
c)^6 + (a*d*cos(d*x + c)^8 - a*d*cos(d*x + c)^6)*sin(d*x + c))

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giac [A]  time = 0.30, size = 182, normalized size = 0.78 \[ -\frac {\frac {11460 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3900 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {15360 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {1536 \, {\left (15 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )^{2}} - \frac {2 \, {\left (3575 \, \sin \left (d x + c\right )^{3} - 11553 \, \sin \left (d x + c\right )^{2} + 12513 \, \sin \left (d x + c\right ) - 4551\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {23875 \, \sin \left (d x + c\right )^{4} + 101644 \, \sin \left (d x + c\right )^{3} + 163074 \, \sin \left (d x + c\right )^{2} + 117036 \, \sin \left (d x + c\right ) + 31779}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/3072*(11460*log(abs(sin(d*x + c) + 1))/a + 3900*log(abs(sin(d*x + c) - 1))/a - 15360*log(abs(sin(d*x + c)))
/a + 1536*(15*sin(d*x + c)^2 - 2*sin(d*x + c) + 1)/(a*sin(d*x + c)^2) - 2*(3575*sin(d*x + c)^3 - 11553*sin(d*x
+ c)^2 + 12513*sin(d*x + c) - 4551)/(a*(sin(d*x + c) - 1)^3) - (23875*sin(d*x + c)^4 + 101644*sin(d*x + c)^3
+ 163074*sin(d*x + c)^2 + 117036*sin(d*x + c) + 31779)/(a*(sin(d*x + c) + 1)^4))/d

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maple [A]  time = 0.53, size = 208, normalized size = 0.90 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {11}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {69}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{2 a d \sin \left (d x +c \right )^{2}}+\frac {1}{d a \sin \left (d x +c \right )}+\frac {5 \ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{48 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {29}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {2}{a d \left (1+\sin \left (d x +c \right )\right )}-\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x)

[Out]

-1/96/a/d/(sin(d*x+c)-1)^3+11/128/a/d/(sin(d*x+c)-1)^2-69/128/a/d/(sin(d*x+c)-1)-325/256/a/d*ln(sin(d*x+c)-1)-
1/2/a/d/sin(d*x+c)^2+1/d/a/sin(d*x+c)+5*ln(sin(d*x+c))/a/d+1/64/a/d/(1+sin(d*x+c))^4+5/48/a/d/(1+sin(d*x+c))^3
+29/64/a/d/(1+sin(d*x+c))^2+2/a/d/(1+sin(d*x+c))-955/256*ln(1+sin(d*x+c))/a/d

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maxima [A]  time = 0.34, size = 217, normalized size = 0.94 \[ \frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{8} - 15 \, \sin \left (d x + c\right )^{7} - 3480 \, \sin \left (d x + c\right )^{6} - 120 \, \sin \left (d x + c\right )^{5} + 4479 \, \sin \left (d x + c\right )^{4} + 319 \, \sin \left (d x + c\right )^{3} - 2192 \, \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right ) + 192\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 3 \, a \sin \left (d x + c\right )^{7} - 3 \, a \sin \left (d x + c\right )^{6} + 3 \, a \sin \left (d x + c\right )^{5} + 3 \, a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2}} - \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {3840 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/768*(2*(945*sin(d*x + c)^8 - 15*sin(d*x + c)^7 - 3480*sin(d*x + c)^6 - 120*sin(d*x + c)^5 + 4479*sin(d*x + c
)^4 + 319*sin(d*x + c)^3 - 2192*sin(d*x + c)^2 - 192*sin(d*x + c) + 192)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8
- 3*a*sin(d*x + c)^7 - 3*a*sin(d*x + c)^6 + 3*a*sin(d*x + c)^5 + 3*a*sin(d*x + c)^4 - a*sin(d*x + c)^3 - a*sin
(d*x + c)^2) - 2865*log(sin(d*x + c) + 1)/a - 975*log(sin(d*x + c) - 1)/a + 3840*log(sin(d*x + c))/a)/d

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mupad [B]  time = 9.24, size = 223, normalized size = 0.96 \[ \frac {5\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {955\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {325\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {-\frac {315\,{\sin \left (c+d\,x\right )}^8}{128}+\frac {5\,{\sin \left (c+d\,x\right )}^7}{128}+\frac {145\,{\sin \left (c+d\,x\right )}^6}{16}+\frac {5\,{\sin \left (c+d\,x\right )}^5}{16}-\frac {1493\,{\sin \left (c+d\,x\right )}^4}{128}-\frac {319\,{\sin \left (c+d\,x\right )}^3}{384}+\frac {137\,{\sin \left (c+d\,x\right )}^2}{24}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^9-a\,{\sin \left (c+d\,x\right )}^8+3\,a\,{\sin \left (c+d\,x\right )}^7+3\,a\,{\sin \left (c+d\,x\right )}^6-3\,a\,{\sin \left (c+d\,x\right )}^5-3\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^7*sin(c + d*x)^3*(a + a*sin(c + d*x))),x)

[Out]

(5*log(sin(c + d*x)))/(a*d) - (955*log(sin(c + d*x) + 1))/(256*a*d) - (325*log(sin(c + d*x) - 1))/(256*a*d) +
(sin(c + d*x)/2 + (137*sin(c + d*x)^2)/24 - (319*sin(c + d*x)^3)/384 - (1493*sin(c + d*x)^4)/128 + (5*sin(c +
d*x)^5)/16 + (145*sin(c + d*x)^6)/16 + (5*sin(c + d*x)^7)/128 - (315*sin(c + d*x)^8)/128 - 1/2)/(d*(a*sin(c +
d*x)^2 + a*sin(c + d*x)^3 - 3*a*sin(c + d*x)^4 - 3*a*sin(c + d*x)^5 + 3*a*sin(c + d*x)^6 + 3*a*sin(c + d*x)^7
- a*sin(c + d*x)^8 - a*sin(c + d*x)^9))

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

[Out]

Timed out

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