\(\int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx\) [46]

Optimal result

Integrand size = 21, antiderivative size = 157 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]

[Out]

-1/6*a^6/d/(a-a*cos(d*x+c))^3-7/8*a^5/d/(a-a*cos(d*x+c))^2-31/8*a^4/d/(a-a*cos(d*x+c))+111/16*a^3*ln(1-cos(d*x
+c))/d-7*a^3*ln(cos(d*x+c))/d+1/16*a^3*ln(1+cos(d*x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (\cos (c+d x)+1)}{16 d} \]

[In]

Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]

[Out]

-1/6*a^6/(d*(a - a*Cos[c + d*x])^3) - (7*a^5)/(8*d*(a - a*Cos[c + d*x])^2) - (31*a^4)/(8*d*(a - a*Cos[c + d*x]
)) + (111*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (7*a^3*Log[Cos[c + d*x]])/d + (a^3*Log[1 + Cos[c + d*x]])/(16*d)
+ (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

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]))

Rule 2915

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/b)*x
)^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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^7(c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a^7 \text {Subst}\left (\int \frac {a^3}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^{10} \text {Subst}\left (\int \frac {1}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^{10} \text {Subst}\left (\int \left (-\frac {1}{16 a^7 (a-x)}-\frac {1}{a^5 x^3}+\frac {3}{a^6 x^2}-\frac {7}{a^7 x}+\frac {1}{2 a^4 (a+x)^4}+\frac {7}{4 a^5 (a+x)^3}+\frac {31}{8 a^6 (a+x)^2}+\frac {111}{16 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (186 \csc ^2\left (\frac {1}{2} (c+d x)\right )+21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+2 \csc ^6\left (\frac {1}{2} (c+d x)\right )-12 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-56 \log (\cos (c+d x))+111 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \sec (c+d x)+4 \sec ^2(c+d x)\right )\right )}{768 d} \]

[In]

Integrate[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]

[Out]

-1/768*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(186*Csc[(c + d*x)/2]^2 + 21*Csc[(c + d*x)/2]^4 + 2*Csc[(c
+ d*x)/2]^6 - 12*(Log[Cos[(c + d*x)/2]] - 56*Log[Cos[c + d*x]] + 111*Log[Sin[(c + d*x)/2]] + 24*Sec[c + d*x]
+ 4*Sec[c + d*x]^2)))/d

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\frac {79 \left (\frac {112 \left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{79}+\frac {112 \left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{79}+\frac {222 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{79}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\cos \left (d x +c \right )-\frac {161 \cos \left (2 d x +2 c \right )}{237}+\frac {71 \cos \left (3 d x +3 c \right )}{237}-\frac {449 \cos \left (4 d x +4 c \right )}{7584}-\frac {4319}{7584}\right )\right ) a^{3}}{16 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(164\)
norman	\(\frac {-\frac {a^{3}}{48 d}-\frac {23 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{96 d}-\frac {91 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{48 d}-\frac {103 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 d}+\frac {339 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {111 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(172\)
risch	\(\frac {a^{3} \left (165 \,{\mathrm e}^{9 i \left (d x +c \right )}-822 \,{\mathrm e}^{8 i \left (d x +c \right )}+1852 \,{\mathrm e}^{7 i \left (d x +c \right )}-2754 \,{\mathrm e}^{6 i \left (d x +c \right )}+3182 \,{\mathrm e}^{5 i \left (d x +c \right )}-2754 \,{\mathrm e}^{4 i \left (d x +c \right )}+1852 \,{\mathrm e}^{3 i \left (d x +c \right )}-822 \,{\mathrm e}^{2 i \left (d x +c \right )}+165 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {111 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(196\)
derivativedivides	\(\frac {a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )^{2}}+4 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\)	\(273\)
default	\(\frac {a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )^{2}}+4 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\)	\(273\)

[In]

int(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

79/16*(112/79*(-cos(2*d*x+2*c)-1)*ln(tan(1/2*d*x+1/2*c)-1)+112/79*(-cos(2*d*x+2*c)-1)*ln(tan(1/2*d*x+1/2*c)+1)
+222/79*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c))+cot(1/2*d*x+1/2*c)^2*csc(1/2*d*x+1/2*c)^4*(cos(d*x+c)-161/23
7*cos(2*d*x+2*c)+71/237*cos(3*d*x+3*c)-449/7584*cos(4*d*x+4*c)-4319/7584))*a^3/d/(1+cos(2*d*x+2*c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (148) = 296\).

Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.89 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {330 \, a^{3} \cos \left (d x + c\right )^{4} - 822 \, a^{3} \cos \left (d x + c\right )^{3} + 596 \, a^{3} \cos \left (d x + c\right )^{2} - 72 \, a^{3} \cos \left (d x + c\right ) - 24 \, a^{3} - 336 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 333 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/48*(330*a^3*cos(d*x + c)^4 - 822*a^3*cos(d*x + c)^3 + 596*a^3*cos(d*x + c)^2 - 72*a^3*cos(d*x + c) - 24*a^3
- 336*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^3 - a^3*cos(d*x + c)^2)*log(-cos(d*x + c
)) + 3*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^3 - a^3*cos(d*x + c)^2)*log(1/2*cos(d*x
+ c) + 1/2) + 333*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^3 - a^3*cos(d*x + c)^2)*log
(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 - d*cos(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**7*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 333 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 336 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (165 \, a^{3} \cos \left (d x + c\right )^{4} - 411 \, a^{3} \cos \left (d x + c\right )^{3} + 298 \, a^{3} \cos \left (d x + c\right )^{2} - 36 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{48 \, d} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/48*(3*a^3*log(cos(d*x + c) + 1) + 333*a^3*log(cos(d*x + c) - 1) - 336*a^3*log(cos(d*x + c)) + 2*(165*a^3*cos
(d*x + c)^4 - 411*a^3*cos(d*x + c)^3 + 298*a^3*cos(d*x + c)^2 - 36*a^3*cos(d*x + c) - 12*a^3)/(cos(d*x + c)^5
- 3*cos(d*x + c)^4 + 3*cos(d*x + c)^3 - cos(d*x + c)^2))/d

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.55 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {666 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 672 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (2 \, a^{3} - \frac {27 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {234 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1221 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} + \frac {48 \, {\left (33 \, a^{3} + \frac {50 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{96 \, d} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/96*(666*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 672*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) - 1)) + (2*a^3 - 27*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 234*a^3*(cos(d*x + c) - 1)^2/(cos(d
*x + c) + 1)^2 - 1221*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3
+ 48*(33*a^3 + 50*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 21*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^
2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2)/d

Mupad [B] (verification not implemented)

Time = 13.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {111\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}+\frac {a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {-\frac {55\,a^3\,{\cos \left (c+d\,x\right )}^4}{8}+\frac {137\,a^3\,{\cos \left (c+d\,x\right )}^3}{8}-\frac {149\,a^3\,{\cos \left (c+d\,x\right )}^2}{12}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left (-{\cos \left (c+d\,x\right )}^5+3\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {7\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]

[In]

int((a + a/cos(c + d*x))^3/sin(c + d*x)^7,x)

[Out]

(111*a^3*log(cos(c + d*x) - 1))/(16*d) + (a^3*log(cos(c + d*x) + 1))/(16*d) + ((3*a^3*cos(c + d*x))/2 + a^3/2
- (149*a^3*cos(c + d*x)^2)/12 + (137*a^3*cos(c + d*x)^3)/8 - (55*a^3*cos(c + d*x)^4)/8)/(d*(cos(c + d*x)^2 - 3
*cos(c + d*x)^3 + 3*cos(c + d*x)^4 - cos(c + d*x)^5)) - (7*a^3*log(cos(c + d*x)))/d