3.891 \(\int \frac{\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=217 \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{12 d (a \sin (c+d x)+a)^3}+\frac{9 a}{128 d (a-a \sin (c+d x))^2}-\frac{19 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}-\frac{35}{32 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{443 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

-(Csc[c + d*x]/(a*d)) - (187*Log[1 - Sin[c + d*x]])/(256*a*d) - Log[Sin[c + d*x]]/(a*d) + (443*Log[1 + Sin[c +
d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (9*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a -
a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(12*d*(a + a*Sin[c + d*x])^3) - (19*a)/(64*d*(a + a
*Sin[c + d*x])^2) - 35/(32*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.239394, antiderivative size = 217, 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} \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{12 d (a \sin (c+d x)+a)^3}+\frac{9 a}{128 d (a-a \sin (c+d x))^2}-\frac{19 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}-\frac{35}{32 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{443 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x]^2*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

-(Csc[c + d*x]/(a*d)) - (187*Log[1 - Sin[c + d*x]])/(256*a*d) - Log[Sin[c + d*x]]/(a*d) + (443*Log[1 + Sin[c +
d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (9*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a -
a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(12*d*(a + a*Sin[c + d*x])^3) - (19*a)/(64*d*(a + a
*Sin[c + d*x])^2) - 35/(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 ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{a^2}{(a-x)^4 x^2 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \frac{1}{(a-x)^4 x^2 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \left (\frac{1}{32 a^7 (a-x)^4}+\frac{9}{64 a^8 (a-x)^3}+\frac{47}{128 a^9 (a-x)^2}+\frac{187}{256 a^{10} (a-x)}+\frac{1}{a^9 x^2}-\frac{1}{a^{10} x}+\frac{1}{16 a^6 (a+x)^5}+\frac{1}{4 a^7 (a+x)^4}+\frac{19}{32 a^8 (a+x)^3}+\frac{35}{32 a^9 (a+x)^2}+\frac{443}{256 a^{10} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{443 \log (1+\sin (c+d x))}{256 a d}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{9 a}{128 d (a-a \sin (c+d x))^2}+\frac{47}{128 d (a-a \sin (c+d x))}-\frac{a^3}{64 d (a+a \sin (c+d x))^4}-\frac{a^2}{12 d (a+a \sin (c+d x))^3}-\frac{19 a}{64 d (a+a \sin (c+d x))^2}-\frac{35}{32 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.13852, size = 201, normalized size = 0.93 \[ \frac{a^9 \left (\frac{47}{128 a^9 (a-a \sin (c+d x))}-\frac{35}{32 a^9 (a \sin (c+d x)+a)}+\frac{9}{128 a^8 (a-a \sin (c+d x))^2}-\frac{19}{64 a^8 (a \sin (c+d x)+a)^2}+\frac{1}{96 a^7 (a-a \sin (c+d x))^3}-\frac{1}{12 a^7 (a \sin (c+d x)+a)^3}-\frac{1}{64 a^6 (a \sin (c+d x)+a)^4}-\frac{\csc (c+d x)}{a^{10}}-\frac{187 \log (1-\sin (c+d x))}{256 a^{10}}-\frac{\log (\sin (c+d x))}{a^{10}}+\frac{443 \log (\sin (c+d x)+1)}{256 a^{10}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(a^9*(-(Csc[c + d*x]/a^10) - (187*Log[1 - Sin[c + d*x]])/(256*a^10) - Log[Sin[c + d*x]]/a^10 + (443*Log[1 + Si
n[c + d*x]])/(256*a^10) + 1/(96*a^7*(a - a*Sin[c + d*x])^3) + 9/(128*a^8*(a - a*Sin[c + d*x])^2) + 47/(128*a^9
*(a - a*Sin[c + d*x])) - 1/(64*a^6*(a + a*Sin[c + d*x])^4) - 1/(12*a^7*(a + a*Sin[c + d*x])^3) - 19/(64*a^8*(a
+ a*Sin[c + d*x])^2) - 35/(32*a^9*(a + a*Sin[c + d*x]))))/d

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Maple [A]  time = 0.103, size = 193, normalized size = 0.9 \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{9}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{47}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{187\,\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}{12\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{19}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{35}{32\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{443\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}}-{\frac{1}{da\sin \left ( dx+c \right ) }}-{\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)^2*sec(d*x+c)^7/(a+a*sin(d*x+c)),x)

[Out]

-1/96/d/a/(sin(d*x+c)-1)^3+9/128/d/a/(sin(d*x+c)-1)^2-47/128/a/d/(sin(d*x+c)-1)-187/256/a/d*ln(sin(d*x+c)-1)-1
/64/d/a/(1+sin(d*x+c))^4-1/12/d/a/(1+sin(d*x+c))^3-19/64/a/d/(1+sin(d*x+c))^2-35/32/a/d/(1+sin(d*x+c))+443/256
*ln(1+sin(d*x+c))/a/d-1/d/a/sin(d*x+c)-ln(sin(d*x+c))/a/d

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Maxima [A]  time = 1.02794, size = 277, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \,{\left (945 \, \sin \left (d x + c\right )^{7} + 753 \, \sin \left (d x + c\right )^{6} - 2712 \, \sin \left (d x + c\right )^{5} - 2040 \, \sin \left (d x + c\right )^{4} + 2559 \, \sin \left (d x + c\right )^{3} + 1727 \, \sin \left (d x + c\right )^{2} - 784 \, \sin \left (d x + c\right ) - 384\right )}}{a \sin \left (d x + c\right )^{8} + 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} + 3 \, a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right )} - \frac{1329 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{561 \, \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)^2*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(945*sin(d*x + c)^7 + 753*sin(d*x + c)^6 - 2712*sin(d*x + c)^5 - 2040*sin(d*x + c)^4 + 2559*sin(d*x
+ c)^3 + 1727*sin(d*x + c)^2 - 784*sin(d*x + c) - 384)/(a*sin(d*x + c)^8 + 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 + 3*a*sin(d*x + c)^3 - a*sin(d*x + c)^2 - a*sin(d*x + c)) - 1329*
log(sin(d*x + c) + 1)/a + 561*log(sin(d*x + c) - 1)/a + 768*log(sin(d*x + c))/a)/d

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Fricas [A]  time = 1.67313, size = 694, normalized size = 3.2 \begin{align*} \frac{1506 \, \cos \left (d x + c\right )^{6} - 438 \, \cos \left (d x + c\right )^{4} - 188 \, \cos \left (d x + c\right )^{2} - 768 \,{\left (\cos \left (d x + c\right )^{8} - \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 ) + 1329 \,{\left (\cos \left (d x + c\right )^{8} - \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 ) - 561 \,{\left (\cos \left (d x + c\right )^{8} - \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 (945 \, \cos \left (d x + c\right )^{6} - 123 \, \cos \left (d x + c\right )^{4} - 30 \, \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} \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)^2*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(1506*cos(d*x + c)^6 - 438*cos(d*x + c)^4 - 188*cos(d*x + c)^2 - 768*(cos(d*x + c)^8 - cos(d*x + c)^6*si
n(d*x + c) - cos(d*x + c)^6)*log(1/2*sin(d*x + c)) + 1329*(cos(d*x + c)^8 - cos(d*x + c)^6*sin(d*x + c) - cos(
d*x + c)^6)*log(sin(d*x + c) + 1) - 561*(cos(d*x + c)^8 - cos(d*x + c)^6*sin(d*x + c) - cos(d*x + c)^6)*log(-s
in(d*x + c) + 1) + 2*(945*cos(d*x + c)^6 - 123*cos(d*x + c)^4 - 30*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*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(csc(d*x+c)**2*sec(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.36998, size = 230, normalized size = 1.06 \begin{align*} \frac{\frac{5316 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{2244 \, \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{3072 \,{\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac{2 \,{\left (2057 \, \sin \left (d x + c\right )^{3} - 6735 \, \sin \left (d x + c\right )^{2} + 7407 \, \sin \left (d x + c\right ) - 2745\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{11075 \, \sin \left (d x + c\right )^{4} + 47660 \, \sin \left (d x + c\right )^{3} + 77442 \, \sin \left (d x + c\right )^{2} + 56460 \, \sin \left (d x + c\right ) + 15651}{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)^2*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/3072*(5316*log(abs(sin(d*x + c) + 1))/a - 2244*log(abs(sin(d*x + c) - 1))/a - 3072*log(abs(sin(d*x + c)))/a
+ 3072*(sin(d*x + c) - 1)/(a*sin(d*x + c)) + 2*(2057*sin(d*x + c)^3 - 6735*sin(d*x + c)^2 + 7407*sin(d*x + c)
- 2745)/(a*(sin(d*x + c) - 1)^3) - (11075*sin(d*x + c)^4 + 47660*sin(d*x + c)^3 + 77442*sin(d*x + c)^2 + 56460
*sin(d*x + c) + 15651)/(a*(sin(d*x + c) + 1)^4))/d