3.893 \(\int \frac{\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=253 \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{8 d (a \sin (c+d x)+a)^3}+\frac{13 a}{128 d (a-a \sin (c+d x))^2}-\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}-\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{5 \csc (c+d x)}{a d}-\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{5 \log (\sin (c+d x))}{a d}+\frac{1795 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

(-5*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - (515*Log[1 - Sin[c + d*x]])/(256*a
*d) - (5*Log[Sin[c + d*x]])/(a*d) + (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3)
+ (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4)
- a^2/(8*d*(a + a*Sin[c + d*x])^3) - (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 105/(32*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.264146, antiderivative size = 253, 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} \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{8 d (a \sin (c+d x)+a)^3}+\frac{13 a}{128 d (a-a \sin (c+d x))^2}-\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}-\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{5 \csc (c+d x)}{a d}-\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{5 \log (\sin (c+d x))}{a d}+\frac{1795 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - (515*Log[1 - Sin[c + d*x]])/(256*a
*d) - (5*Log[Sin[c + d*x]])/(a*d) + (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3)
+ (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4)
- a^2/(8*d*(a + a*Sin[c + d*x])^3) - (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 105/(32*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 ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{a^4}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \frac{1}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \left (\frac{1}{32 a^9 (a-x)^4}+\frac{13}{64 a^{10} (a-x)^3}+\frac{95}{128 a^{11} (a-x)^2}+\frac{515}{256 a^{12} (a-x)}+\frac{1}{a^9 x^4}-\frac{1}{a^{10} x^3}+\frac{5}{a^{11} x^2}-\frac{5}{a^{12} x}+\frac{1}{16 a^8 (a+x)^5}+\frac{3}{8 a^9 (a+x)^4}+\frac{41}{32 a^{10} (a+x)^3}+\frac{105}{32 a^{11} (a+x)^2}+\frac{1795}{256 a^{12} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{5 \csc (c+d x)}{a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}-\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{5 \log (\sin (c+d x))}{a d}+\frac{1795 \log (1+\sin (c+d x))}{256 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{95}{128 d (a-a \sin (c+d x))}-\frac{a^3}{64 d (a+a \sin (c+d x))^4}-\frac{a^2}{8 d (a+a \sin (c+d x))^3}-\frac{41 a}{64 d (a+a \sin (c+d x))^2}-\frac{105}{32 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.13887, size = 231, normalized size = 0.91 \[ \frac{a^{11} \left (\frac{95}{128 a^{11} (a-a \sin (c+d x))}-\frac{105}{32 a^{11} (a \sin (c+d x)+a)}+\frac{13}{128 a^{10} (a-a \sin (c+d x))^2}-\frac{41}{64 a^{10} (a \sin (c+d x)+a)^2}+\frac{1}{96 a^9 (a-a \sin (c+d x))^3}-\frac{1}{8 a^9 (a \sin (c+d x)+a)^3}-\frac{1}{64 a^8 (a \sin (c+d x)+a)^4}-\frac{\csc ^3(c+d x)}{3 a^{12}}+\frac{\csc ^2(c+d x)}{2 a^{12}}-\frac{5 \csc (c+d x)}{a^{12}}-\frac{515 \log (1-\sin (c+d x))}{256 a^{12}}-\frac{5 \log (\sin (c+d x))}{a^{12}}+\frac{1795 \log (\sin (c+d x)+1)}{256 a^{12}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^11*((-5*Csc[c + d*x])/a^12 + Csc[c + d*x]^2/(2*a^12) - Csc[c + d*x]^3/(3*a^12) - (515*Log[1 - Sin[c + d*x]]
)/(256*a^12) - (5*Log[Sin[c + d*x]])/a^12 + (1795*Log[1 + Sin[c + d*x]])/(256*a^12) + 1/(96*a^9*(a - a*Sin[c +
d*x])^3) + 13/(128*a^10*(a - a*Sin[c + d*x])^2) + 95/(128*a^11*(a - a*Sin[c + d*x])) - 1/(64*a^8*(a + a*Sin[c
+ d*x])^4) - 1/(8*a^9*(a + a*Sin[c + d*x])^3) - 41/(64*a^10*(a + a*Sin[c + d*x])^2) - 105/(32*a^11*(a + a*Sin
[c + d*x]))))/d

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Maple [A]  time = 0.114, size = 225, normalized size = 0.9 \begin{align*} -{\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{95}{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{1}{8\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{41}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{105}{32\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{1795\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-5\,{\frac{1}{da\sin \left ( dx+c \right ) }}-5\,{\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)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x)

[Out]

-1/96/d/a/(sin(d*x+c)-1)^3+13/128/d/a/(sin(d*x+c)-1)^2-95/128/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))^4-1/8/d/a/(1+sin(d*x+c))^3-41/64/a/d/(1+sin(d*x+c))^2-105/32/a/d/(1+sin(d*x+c))+1795/2
56*ln(1+sin(d*x+c))/a/d-1/3/d/a/sin(d*x+c)^3+1/2/d/a/sin(d*x+c)^2-5/d/a/sin(d*x+c)-5*ln(sin(d*x+c))/a/d

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Maxima [A]  time = 1.06219, size = 306, normalized size = 1.21 \begin{align*} -\frac{\frac{2 \,{\left (3465 \, \sin \left (d x + c\right )^{9} + 2505 \, \sin \left (d x + c\right )^{8} - 10200 \, \sin \left (d x + c\right )^{7} - 6840 \, \sin \left (d x + c\right )^{6} + 10023 \, \sin \left (d x + c\right )^{5} + 5863 \, \sin \left (d x + c\right )^{4} - 3344 \, \sin \left (d x + c\right )^{3} - 1344 \, \sin \left (d x + c\right )^{2} + 64 \, \sin \left (d x + c\right ) - 128\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 3 \, 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} - a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3}} - \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} + \frac{3840 \, \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)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(3465*sin(d*x + c)^9 + 2505*sin(d*x + c)^8 - 10200*sin(d*x + c)^7 - 6840*sin(d*x + c)^6 + 10023*sin(
d*x + c)^5 + 5863*sin(d*x + c)^4 - 3344*sin(d*x + c)^3 - 1344*sin(d*x + c)^2 + 64*sin(d*x + c) - 128)/(a*sin(d
*x + c)^10 + a*sin(d*x + c)^9 - 3*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 - a*sin(d*x + c)^4 - a*sin(d*x + c)^3) - 5385*log(sin(d*x + c) + 1)/a + 1545*log(sin(d*x + c) - 1)/a + 384
0*log(sin(d*x + c))/a)/d

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Fricas [A]  time = 1.70826, size = 979, normalized size = 3.87 \begin{align*} \frac{5010 \, \cos \left (d x + c\right )^{8} - 6360 \, \cos \left (d x + c\right )^{6} + 746 \, \cos \left (d x + c\right )^{4} + 236 \, \cos \left (d x + c\right )^{2} - 3840 \,{\left (\cos \left (d x + c\right )^{10} - 2 \, \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 ) + 5385 \,{\left (\cos \left (d x + c\right )^{10} - 2 \, \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 ) - 1545 \,{\left (\cos \left (d x + c\right )^{10} - 2 \, \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 (3465 \, \cos \left (d x + c\right )^{8} - 3660 \, \cos \left (d x + c\right )^{6} + 213 \, \cos \left (d x + c\right )^{4} + 38 \, \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 )^{10} - 2 \, 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(5010*cos(d*x + c)^8 - 6360*cos(d*x + c)^6 + 746*cos(d*x + c)^4 + 236*cos(d*x + c)^2 - 3840*(cos(d*x + c
)^10 - 2*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
)) + 5385*(cos(d*x + c)^10 - 2*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) - 1545*(cos(d*x + c)^10 - 2*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*(3465*cos(d*x + c)^8 - 3660*cos(d*x + c)^6 + 213*cos(d*x +
c)^4 + 38*cos(d*x + c)^2 + 8)*sin(d*x + c) + 112)/(a*d*cos(d*x + c)^10 - 2*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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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)**4*sec(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.39332, size = 252, normalized size = 1. \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{15360 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{19745 \, \sin \left (d x + c\right )^{6} - 76875 \, \sin \left (d x + c\right )^{5} + 111723 \, \sin \left (d x + c\right )^{4} - 74081 \, \sin \left (d x + c\right )^{3} + 23040 \, \sin \left (d x + c\right )^{2} - 4608 \, \sin \left (d x + c\right ) + 1024}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{3} a} - \frac{44875 \, \sin \left (d x + c\right )^{4} + 189580 \, \sin \left (d x + c\right )^{3} + 301458 \, \sin \left (d x + c\right )^{2} + 214060 \, \sin \left (d x + c\right ) + 57355}{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)^4*sec(d*x+c)^7/(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 - 15360*log(abs(sin(d*x + c)))/
a + (19745*sin(d*x + c)^6 - 76875*sin(d*x + c)^5 + 111723*sin(d*x + c)^4 - 74081*sin(d*x + c)^3 + 23040*sin(d*
x + c)^2 - 4608*sin(d*x + c) + 1024)/((sin(d*x + c)^2 - sin(d*x + c))^3*a) - (44875*sin(d*x + c)^4 + 189580*si
n(d*x + c)^3 + 301458*sin(d*x + c)^2 + 214060*sin(d*x + c) + 57355)/(a*(sin(d*x + c) + 1)^4))/d