3.859 \(\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=143 \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d} \]

[Out]

-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (39*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]
])/d - (9*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(d*(a - a*Sin[c + d*x])) +
a^2/(8*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.132132, antiderivative size = 143, normalized size of antiderivative = 1., 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} \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (39*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]
])/d - (9*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(d*(a - a*Sin[c + d*x])) +
a^2/(8*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 \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{a^3}{(a-x)^3 x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{1}{4 a^5 (a-x)^3}+\frac{1}{a^6 (a-x)^2}+\frac{39}{16 a^7 (a-x)}+\frac{1}{a^5 x^3}+\frac{1}{a^6 x^2}+\frac{3}{a^7 x}-\frac{1}{8 a^6 (a+x)^2}-\frac{9}{16 a^7 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{9 a \log (1+\sin (c+d x))}{16 d}+\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [C]  time = 0.753603, size = 86, normalized size = 0.6 \[ -\frac{a \csc (c+d x) \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\sin ^2(c+d x)\right )}{d}-\frac{a \left (2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-((a*Csc[c + d*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[c + d*x]^2])/d) - (a*(2*Csc[c + d*x]^2 + 12*Log[Cos[c +
d*x]] - 12*Log[Sin[c + d*x]] - 4*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)

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Maple [A]  time = 0.112, size = 151, normalized size = 1.1 \begin{align*}{\frac{a}{4\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{5\,a}{8\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{15\,a}{8\,d\sin \left ( dx+c \right ) }}+{\frac{15\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{a}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{a\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c)),x)

[Out]

1/4/d*a/sin(d*x+c)/cos(d*x+c)^4+5/8/d*a/sin(d*x+c)/cos(d*x+c)^2-15/8/d*a/sin(d*x+c)+15/8/d*a*ln(sec(d*x+c)+tan
(d*x+c))+1/4/d*a/sin(d*x+c)^2/cos(d*x+c)^4+3/4/d*a/sin(d*x+c)^2/cos(d*x+c)^2-3/2/d*a/sin(d*x+c)^2+3*a*ln(tan(d
*x+c))/d

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Maxima [A]  time = 1.13342, size = 171, normalized size = 1.2 \begin{align*} -\frac{9 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) + 39 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - 48 \, a \log \left (\sin \left (d x + c\right )\right ) + \frac{2 \,{\left (15 \, a \sin \left (d x + c\right )^{4} - 3 \, a \sin \left (d x + c\right )^{3} - 22 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 4 \, a\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} - \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/16*(9*a*log(sin(d*x + c) + 1) + 39*a*log(sin(d*x + c) - 1) - 48*a*log(sin(d*x + c)) + 2*(15*a*sin(d*x + c)^
4 - 3*a*sin(d*x + c)^3 - 22*a*sin(d*x + c)^2 + 4*a*sin(d*x + c) + 4*a)/(sin(d*x + c)^5 - sin(d*x + c)^4 - sin(
d*x + c)^3 + sin(d*x + c)^2))/d

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Fricas [B]  time = 1.58582, size = 733, normalized size = 5.13 \begin{align*} \frac{30 \, a \cos \left (d x + c\right )^{4} - 16 \, a \cos \left (d x + c\right )^{2} + 48 \,{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 9 \,{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 39 \,{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) - 6 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2} -{\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\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)^3*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/16*(30*a*cos(d*x + c)^4 - 16*a*cos(d*x + c)^2 + 48*(a*cos(d*x + c)^4 - a*cos(d*x + c)^2 - (a*cos(d*x + c)^4
- a*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*sin(d*x + c)) - 9*(a*cos(d*x + c)^4 - a*cos(d*x + c)^2 - (a*cos(d*x
+ c)^4 - a*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 39*(a*cos(d*x + c)^4 - a*cos(d*x + c)^2 - (a*
cos(d*x + c)^4 - a*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(3*a*cos(d*x + c)^2 + a)*sin(d*x +
c) - 6*a)/(d*cos(d*x + c)^4 - d*cos(d*x + c)^2 - (d*cos(d*x + c)^4 - d*cos(d*x + c)^2)*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)**3*sec(d*x+c)**5*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.34488, size = 169, normalized size = 1.18 \begin{align*} -\frac{36 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 156 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 192 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{4 \,{\left (9 \, a \sin \left (d x + c\right ) + 11 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac{27 \, a \sin \left (d x + c\right )^{4} + 74 \, a \sin \left (d x + c\right )^{3} - 141 \, a \sin \left (d x + c\right )^{2} + 32 \, a}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{2}}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/64*(36*a*log(abs(sin(d*x + c) + 1)) + 156*a*log(abs(sin(d*x + c) - 1)) - 192*a*log(abs(sin(d*x + c))) - 4*(
9*a*sin(d*x + c) + 11*a)/(sin(d*x + c) + 1) + (27*a*sin(d*x + c)^4 + 74*a*sin(d*x + c)^3 - 141*a*sin(d*x + c)^
2 + 32*a)/(sin(d*x + c)^2 - sin(d*x + c))^2)/d