3.879 \(\int \frac{\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=220 \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{6 d (a \sin (c+d x)+a)^3}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{15 a}{128 d (a-a \sin (c+d x))^2}-\frac{55 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}+\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}+\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{955 \log (\sin (c+d x)+1)}{256 a d} \]

[Out]

(325*Log[1 - Sin[c + d*x]])/(256*a*d) + (955*Log[1 + Sin[c + d*x]])/(256*a*d) - Sin[c + d*x]/(a*d) + Sin[c + d
*x]^2/(2*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (15*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*S
in[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(6*d*(a + a*Sin[c + d*x])^3) - (55*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 = 0.228588, antiderivative size = 220, 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} \[ -\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{6 d (a \sin (c+d x)+a)^3}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{15 a}{128 d (a-a \sin (c+d x))^2}-\frac{55 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}+\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}+\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{955 \log (\sin (c+d x)+1)}{256 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(325*Log[1 - Sin[c + d*x]])/(256*a*d) + (955*Log[1 + Sin[c + d*x]])/(256*a*d) - Sin[c + d*x]/(a*d) + Sin[c + d
*x]^2/(2*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (15*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*S
in[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(6*d*(a + a*Sin[c + d*x])^3) - (55*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{\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{x^{10}}{a^{10} (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^{10}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a+\frac{a^5}{32 (a-x)^4}-\frac{15 a^4}{64 (a-x)^3}+\frac{95 a^3}{128 (a-x)^2}-\frac{325 a^2}{256 (a-x)}+x+\frac{a^6}{16 (a+x)^5}-\frac{a^5}{2 (a+x)^4}+\frac{55 a^4}{32 (a+x)^3}-\frac{105 a^3}{32 (a+x)^2}+\frac{955 a^2}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{955 \log (1+\sin (c+d x))}{256 a d}-\frac{\sin (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 a d}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{15 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}{6 d (a+a \sin (c+d x))^3}-\frac{55 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 = 6.14101, size = 143, normalized size = 0.65 \[ \frac{384 \sin ^2(c+d x)-768 \sin (c+d x)+\frac{570}{1-\sin (c+d x)}+\frac{2520}{\sin (c+d x)+1}-\frac{90}{(1-\sin (c+d x))^2}-\frac{660}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}+\frac{128}{(\sin (c+d x)+1)^3}-\frac{12}{(\sin (c+d x)+1)^4}+975 \log (1-\sin (c+d x))+2865 \log (\sin (c+d x)+1)}{768 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(975*Log[1 - Sin[c + d*x]] + 2865*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 90/(1 - Sin[c + d*x])^2 + 5
70/(1 - Sin[c + d*x]) - 768*Sin[c + d*x] + 384*Sin[c + d*x]^2 - 12/(1 + Sin[c + d*x])^4 + 128/(1 + Sin[c + d*x
])^3 - 660/(1 + Sin[c + d*x])^2 + 2520/(1 + Sin[c + d*x]))/(768*a*d)

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Maple [A]  time = 0.112, size = 192, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,da}}-{\frac{\sin \left ( dx+c \right ) }{da}}-{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{15}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{95}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{325\,\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}{6\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{55}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{105}{32\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{955\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x)

[Out]

1/2*sin(d*x+c)^2/d/a-sin(d*x+c)/d/a-1/96/d/a/(sin(d*x+c)-1)^3-15/128/d/a/(sin(d*x+c)-1)^2-95/128/a/d/(sin(d*x+
c)-1)+325/256/a/d*ln(sin(d*x+c)-1)-1/64/d/a/(1+sin(d*x+c))^4+1/6/d/a/(1+sin(d*x+c))^3-55/64/a/d/(1+sin(d*x+c))
^2+105/32/a/d/(1+sin(d*x+c))+955/256*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.03663, size = 266, normalized size = 1.21 \begin{align*} \frac{\frac{2 \,{\left (975 \, \sin \left (d x + c\right )^{6} - 945 \, \sin \left (d x + c\right )^{5} - 3240 \, \sin \left (d x + c\right )^{4} + 1560 \, \sin \left (d x + c\right )^{3} + 3489 \, \sin \left (d x + c\right )^{2} - 671 \, \sin \left (d x + c\right ) - 1232\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{384 \,{\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} + \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}}{768 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/768*(2*(975*sin(d*x + c)^6 - 945*sin(d*x + c)^5 - 3240*sin(d*x + c)^4 + 1560*sin(d*x + c)^3 + 3489*sin(d*x +
c)^2 - 671*sin(d*x + c) - 1232)/(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) + 384*(sin(d*x + c)^2 - 2*sin(d*x + c))/a +
2865*log(sin(d*x + c) + 1)/a + 975*log(sin(d*x + c) - 1)/a)/d

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Fricas [A]  time = 1.89573, size = 554, normalized size = 2.52 \begin{align*} \frac{384 \, \cos \left (d x + c\right )^{8} + 1374 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{4} - 132 \, \cos \left (d x + c\right )^{2} + 2865 \,{\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 ) + 975 \,{\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 (192 \, \cos \left (d x + c\right )^{8} + 288 \, \cos \left (d x + c\right )^{6} - 945 \, \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{2} - 56\right )} \sin \left (d x + c\right ) + 16}{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(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(384*cos(d*x + c)^8 + 1374*cos(d*x + c)^6 + 630*cos(d*x + c)^4 - 132*cos(d*x + c)^2 + 2865*(cos(d*x + c)
^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 975*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*l
og(-sin(d*x + c) + 1) - 2*(192*cos(d*x + c)^8 + 288*cos(d*x + c)^6 - 945*cos(d*x + c)^4 + 330*cos(d*x + c)^2 -
56)*sin(d*x + c) + 16)/(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(sec(d*x+c)**7*sin(d*x+c)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.39147, size = 217, normalized size = 0.99 \begin{align*} \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{1536 \,{\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac{2 \,{\left (3575 \, \sin \left (d x + c\right )^{3} - 9585 \, \sin \left (d x + c\right )^{2} + 8625 \, \sin \left (d x + c\right ) - 2599\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{23875 \, \sin \left (d x + c\right )^{4} + 85420 \, \sin \left (d x + c\right )^{3} + 115650 \, \sin \left (d x + c\right )^{2} + 70028 \, \sin \left (d x + c\right ) + 15971}{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(sec(d*x+c)^7*sin(d*x+c)^10/(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 + 1536*(a*sin(d*x + c)^2 - 2*a*
sin(d*x + c))/a^2 - 2*(3575*sin(d*x + c)^3 - 9585*sin(d*x + c)^2 + 8625*sin(d*x + c) - 2599)/(a*(sin(d*x + c)
- 1)^3) - (23875*sin(d*x + c)^4 + 85420*sin(d*x + c)^3 + 115650*sin(d*x + c)^2 + 70028*sin(d*x + c) + 15971)/(
a*(sin(d*x + c) + 1)^4))/d