3.860 \(\int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=162 \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{59 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}+\frac{11 a \log (\sin (c+d x)+1)}{16 d} \]

[Out]

(-3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (59*a*Log[1 - Sin[c + d*x]])/(16
*d) + (3*a*Log[Sin[c + d*x]])/d + (11*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (5*
a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.140866, antiderivative size = 162, 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} \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{59 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}+\frac{11 a \log (\sin (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(-3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (59*a*Log[1 - Sin[c + d*x]])/(16
*d) + (3*a*Log[Sin[c + d*x]])/d + (11*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (5*
a^2)/(4*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 ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{a^4}{(a-x)^3 x^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 x^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \left (\frac{1}{4 a^6 (a-x)^3}+\frac{5}{4 a^7 (a-x)^2}+\frac{59}{16 a^8 (a-x)}+\frac{1}{a^5 x^4}+\frac{1}{a^6 x^3}+\frac{3}{a^7 x^2}+\frac{3}{a^8 x}+\frac{1}{8 a^7 (a+x)^2}+\frac{11}{16 a^8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{3 a \csc (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{59 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}+\frac{11 a \log (1+\sin (c+d x))}{16 d}+\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.21069, size = 90, normalized size = 0.56 \[ -\frac{a \csc ^3(c+d x) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\sin ^2(c+d x)\right )}{3 d}-\frac{a \left (2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[c + d*x]^2])/(3*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 = 0.123, size = 173, normalized size = 1.1 \begin{align*}{\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}}+{\frac{a}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{7\,a}{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{35\,a}{24\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{35\,a}{8\,d\sin \left ( dx+c \right ) }}+{\frac{35\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c)),x)

[Out]

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+
1/4/d*a/sin(d*x+c)^3/cos(d*x+c)^4-7/12/d*a/sin(d*x+c)^3/cos(d*x+c)^2+35/24/d*a/sin(d*x+c)/cos(d*x+c)^2-35/8/d*
a/sin(d*x+c)+35/8/d*a*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.03477, size = 186, normalized size = 1.15 \begin{align*} \frac{33 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 177 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac{2 \,{\left (105 \, a \sin \left (d x + c\right )^{5} - 69 \, a \sin \left (d x + c\right )^{4} - 106 \, a \sin \left (d x + c\right )^{3} + 52 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 8 \, a\right )}}{\sin \left (d x + c\right )^{6} - \sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{3}}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/48*(33*a*log(sin(d*x + c) + 1) - 177*a*log(sin(d*x + c) - 1) + 144*a*log(sin(d*x + c)) - 2*(105*a*sin(d*x +
c)^5 - 69*a*sin(d*x + c)^4 - 106*a*sin(d*x + c)^3 + 52*a*sin(d*x + c)^2 + 4*a*sin(d*x + c) + 8*a)/(sin(d*x + c
)^6 - sin(d*x + c)^5 - sin(d*x + c)^4 + sin(d*x + c)^3))/d

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Fricas [B]  time = 1.60815, size = 892, normalized size = 5.51 \begin{align*} -\frac{138 \, a \cos \left (d x + c\right )^{4} - 172 \, a \cos \left (d x + c\right )^{2} - 144 \,{\left (a \cos \left (d x + c\right )^{6} - 2 \, 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 ) - 33 \,{\left (a \cos \left (d x + c\right )^{6} - 2 \, 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 ) + 177 \,{\left (a \cos \left (d x + c\right )^{6} - 2 \, 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 (105 \, a \cos \left (d x + c\right )^{4} - 104 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 18 \, a}{48 \,{\left (d \cos \left (d x + c\right )^{6} - 2 \, 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)^4*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/48*(138*a*cos(d*x + c)^4 - 172*a*cos(d*x + c)^2 - 144*(a*cos(d*x + c)^6 - 2*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)) - 33*(a*cos(d*x + c)^6 - 2*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)
+ 177*(a*cos(d*x + c)^6 - 2*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*(105*a*cos(d*x + c)^4 - 104*a*cos(d*x + c)^2 + 3*a)*sin(d*x + c) + 18*a)/(d
*cos(d*x + c)^6 - 2*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)**4*sec(d*x+c)**5*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.37193, size = 201, normalized size = 1.24 \begin{align*} \frac{66 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 354 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 288 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{6 \,{\left (11 \, a \sin \left (d x + c\right ) + 13 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac{3 \,{\left (177 \, a \sin \left (d x + c\right )^{2} - 394 \, a \sin \left (d x + c\right ) + 221 \, a\right )}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac{16 \,{\left (33 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/96*(66*a*log(abs(sin(d*x + c) + 1)) - 354*a*log(abs(sin(d*x + c) - 1)) + 288*a*log(abs(sin(d*x + c))) - 6*(1
1*a*sin(d*x + c) + 13*a)/(sin(d*x + c) + 1) + 3*(177*a*sin(d*x + c)^2 - 394*a*sin(d*x + c) + 221*a)/(sin(d*x +
c) - 1)^2 - 16*(33*a*sin(d*x + c)^3 + 18*a*sin(d*x + c)^2 + 3*a*sin(d*x + c) + 2*a)/sin(d*x + c)^3)/d