3.1.86 \(\int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [86]

Optimal. Leaf size=123 \[ -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{21 a^3 d}+\frac {8 \tan ^3(c+d x)}{63 a^3 d} \]

[Out]

-1/9*sec(d*x+c)^3/d/(a+a*sin(d*x+c))^3-2/21*sec(d*x+c)^3/a/d/(a+a*sin(d*x+c))^2-2/21*sec(d*x+c)^3/d/(a^3+a^3*s
in(d*x+c))+8/21*tan(d*x+c)/a^3/d+8/63*tan(d*x+c)^3/a^3/d

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 3852} \begin {gather*} \frac {8 \tan ^3(c+d x)}{63 a^3 d}+\frac {8 \tan (c+d x)}{21 a^3 d}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {2 \sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]

[Out]

-1/9*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^3) - (2*Sec[c + d*x]^3)/(21*a*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c
+ d*x]^3)/(21*d*(a^3 + a^3*Sin[c + d*x])) + (8*Tan[c + d*x])/(21*a^3*d) + (8*Tan[c + d*x]^3)/(63*a^3*d)

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}+\frac {2 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{3 a}\\ &=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}+\frac {10 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{21 a^2}\\ &=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \int \sec ^4(c+d x) \, dx}{21 a^3}\\ &=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {8 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{21 a^3 d}\\ &=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{21 a^3 d}+\frac {8 \tan ^3(c+d x)}{63 a^3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 85, normalized size = 0.69 \begin {gather*} \frac {\sec ^3(c+d x) (-27 \cos (2 (c+d x))-12 \cos (4 (c+d x))+\cos (6 (c+d x))+36 \sin (c+d x)+2 \sin (3 (c+d x))-6 \sin (5 (c+d x)))}{126 a^3 d (1+\sin (c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(-27*Cos[2*(c + d*x)] - 12*Cos[4*(c + d*x)] + Cos[6*(c + d*x)] + 36*Sin[c + d*x] + 2*Sin[3*(c
+ d*x)] - 6*Sin[5*(c + d*x)]))/(126*a^3*d*(1 + Sin[c + d*x])^3)

________________________________________________________________________________________

Maple [A]
time = 0.33, size = 190, normalized size = 1.54

method	result	size
risch	\(\frac {-\frac {64 \,{\mathrm e}^{i \left (d x +c \right )}}{21}+\frac {64 \,{\mathrm e}^{3 i \left (d x +c \right )}}{63}+\frac {128 \,{\mathrm e}^{5 i \left (d x +c \right )}}{7}-\frac {32 i}{63}+\frac {128 i {\mathrm e}^{2 i \left (d x +c \right )}}{21}+\frac {96 i {\mathrm e}^{4 i \left (d x +c \right )}}{7}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{3}}\)	\(97\)
derivativedivides	\(\frac {-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {46}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d \,a^{3}}\)	\(190\)
default	\(\frac {-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {46}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d \,a^{3}}\)	\(190\)
norman	\(\frac {\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {38}{63 a d}-\frac {6 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {26 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {12 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {34 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21 d a}-\frac {68 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {470 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d a}+\frac {26 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {100 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

2/d/a^3*(-4/9/(tan(1/2*d*x+1/2*c)+1)^9+2/(tan(1/2*d*x+1/2*c)+1)^8-34/7/(tan(1/2*d*x+1/2*c)+1)^7+23/3/(tan(1/2*
d*x+1/2*c)+1)^6-35/4/(tan(1/2*d*x+1/2*c)+1)^5+59/8/(tan(1/2*d*x+1/2*c)+1)^4-19/4/(tan(1/2*d*x+1/2*c)+1)^3+9/4/
(tan(1/2*d*x+1/2*c)+1)^2-57/64/(tan(1/2*d*x+1/2*c)+1)-1/48/(tan(1/2*d*x+1/2*c)-1)^3-1/32/(tan(1/2*d*x+1/2*c)-1
)^2-7/64/(tan(1/2*d*x+1/2*c)-1))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (113) = 226\).
time = 0.30, size = 482, normalized size = 3.92 \begin {gather*} -\frac {2 \, {\left (\frac {51 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {39 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {450 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {306 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {294 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {378 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {273 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {189 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 19\right )}}{63 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-2/63*(51*sin(d*x + c)/(cos(d*x + c) + 1) + 39*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 235*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 - 450*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 306*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 294*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6 + 378*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 63*sin(d*x + c)^8/(cos(d*x + c) + 1
)^8 - 273*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 189*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 63*sin(d*x + c)^11
/(cos(d*x + c) + 1)^11 + 19)/((a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 12*a^3*sin(d*x + c)^2/(cos(d*x +
c) + 1)^2 + 2*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 27*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^3*si
n(d*x + c)^5/(cos(d*x + c) + 1)^5 + 36*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 27*a^3*sin(d*x + c)^8/(cos(d*
x + c) + 1)^8 - 2*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 12*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6*a
^3*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12)*d)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 130, normalized size = 1.06 \begin {gather*} -\frac {16 \, \cos \left (d x + c\right )^{6} - 72 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (24 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 7\right )} \sin \left (d x + c\right ) + 7}{63 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/63*(16*cos(d*x + c)^6 - 72*cos(d*x + c)^4 + 30*cos(d*x + c)^2 - 2*(24*cos(d*x + c)^4 - 20*cos(d*x + c)^2 -
7)*sin(d*x + c) + 7)/(3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*
x + c)^3)*sin(d*x + c))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4/(a+a*sin(d*x+c))**3,x)

[Out]

Integral(sec(c + d*x)**4/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3

________________________________________________________________________________________

Giac [A]
time = 5.15, size = 171, normalized size = 1.39 \begin {gather*} -\frac {\frac {21 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3591 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 19656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 56196 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 95760 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 107730 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 79464 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10944 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1615}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2016*(21*(21*tan(1/2*d*x + 1/2*c)^2 - 36*tan(1/2*d*x + 1/2*c) + 19)/(a^3*(tan(1/2*d*x + 1/2*c) - 1)^3) + (3
591*tan(1/2*d*x + 1/2*c)^8 + 19656*tan(1/2*d*x + 1/2*c)^7 + 56196*tan(1/2*d*x + 1/2*c)^6 + 95760*tan(1/2*d*x +
1/2*c)^5 + 107730*tan(1/2*d*x + 1/2*c)^4 + 79464*tan(1/2*d*x + 1/2*c)^3 + 38484*tan(1/2*d*x + 1/2*c)^2 + 1094
4*tan(1/2*d*x + 1/2*c) + 1615)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d

________________________________________________________________________________________

Mupad [B]
time = 5.56, size = 167, normalized size = 1.36 \begin {gather*} \frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {171\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {145\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {49\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}+\frac {617\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {329\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {145\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {113\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {115\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {19\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{2016\,a^3\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^4*(a + a*sin(c + d*x))^3),x)

[Out]

(cos(c/2 + (d*x)/2)*((63*cos((5*c)/2 + (5*d*x)/2))/8 - (171*cos((3*c)/2 + (3*d*x)/2))/8 - (145*cos((7*c)/2 + (
7*d*x)/2))/16 + (49*cos((9*c)/2 + (9*d*x)/2))/16 + cos((11*c)/2 + (11*d*x)/2)/2 + (617*sin(c/2 + (d*x)/2))/16
- (329*sin((3*c)/2 + (3*d*x)/2))/16 + (145*sin((5*c)/2 + (5*d*x)/2))/32 - (113*sin((7*c)/2 + (7*d*x)/2))/32 -
(115*sin((9*c)/2 + (9*d*x)/2))/32 + (19*sin((11*c)/2 + (11*d*x)/2))/32))/(2016*a^3*d*cos(c/2 - pi/4 + (d*x)/2)
^9*cos(c/2 + pi/4 + (d*x)/2)^3)

________________________________________________________________________________________