\(\int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [73]

Optimal result

Integrand size = 21, antiderivative size = 93 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{7 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{7 a^2 d}+\frac {4 \tan ^3(c+d x)}{21 a^2 d} \]

[Out]

-1/7*sec(d*x+c)^3/d/(a+a*sin(d*x+c))^2-1/7*sec(d*x+c)^3/d/(a^2+a^2*sin(d*x+c))+4/7*tan(d*x+c)/a^2/d+4/21*tan(d
*x+c)^3/a^2/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 3852} \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \tan ^3(c+d x)}{21 a^2 d}+\frac {4 \tan (c+d x)}{7 a^2 d}-\frac {\sec ^3(c+d x)}{7 d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2} \]

[In]

Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/7*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/(7*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*
x])/(7*a^2*d) + (4*Tan[c + d*x]^3)/(21*a^2*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*} \text {integral}& = -\frac {\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}+\frac {5 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{7 a} \\ & = -\frac {\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{7 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \int \sec ^4(c+d x) \, dx}{7 a^2} \\ & = -\frac {\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{7 d \left (a^2+a^2 \sin (c+d x)\right )}-\frac {4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{7 a^2 d} \\ & = -\frac {\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{7 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{7 a^2 d}+\frac {4 \tan ^3(c+d x)}{21 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\sec ^3(c+d x) \left (6-9 \sin (c+d x)-24 \sin ^2(c+d x)-4 \sin ^3(c+d x)+16 \sin ^4(c+d x)+8 \sin ^5(c+d x)\right )}{21 a^2 d (1+\sin (c+d x))^2} \]

[In]

Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/21*(Sec[c + d*x]^3*(6 - 9*Sin[c + d*x] - 24*Sin[c + d*x]^2 - 4*Sin[c + d*x]^3 + 16*Sin[c + d*x]^4 + 8*Sin[c
+ d*x]^5))/(a^2*d*(1 + Sin[c + d*x])^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {16 i \left (8 i {\mathrm e}^{3 i \left (d x +c \right )}+14 \,{\mathrm e}^{4 i \left (d x +c \right )}+4 i {\mathrm e}^{i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{21 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{3} a^{2} d}\)	\(86\)
derivativedivides	\(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {5}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {23}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{2} d}\)	\(158\)
default	\(\frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {5}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {23}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{2} d}\)	\(158\)
parallelrisch	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-76 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+42 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-42 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-21\right )}{21 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\)	\(161\)
norman	\(\frac {\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1}{14 a d}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {20 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{7 d a}-\frac {68 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {13 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {53 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} a}\)	\(209\)

[In]

int(sec(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-16/21*I*(8*I*exp(3*I*(d*x+c))+14*exp(4*I*(d*x+c))+4*I*exp(I*(d*x+c))+3*exp(2*I*(d*x+c))-1)/(exp(I*(d*x+c))+I)
^7/(-I+exp(I*(d*x+c)))^3/a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {16 \, \cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + {\left (8 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 2}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/21*(16*cos(d*x + c)^4 - 8*cos(d*x + c)^2 + (8*cos(d*x + c)^4 - 12*cos(d*x + c)^2 - 5)*sin(d*x + c) - 2)/(a^2
*d*cos(d*x + c)^5 - 2*a^2*d*cos(d*x + c)^3*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^3)

Sympy [F]

\[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

[In]

integrate(sec(d*x+c)**4/(a+a*sin(d*x+c))**2,x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (85) = 170\).

Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.26 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {28 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {42 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {56 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {28 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {42 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {21 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 6\right )}}{21 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-2/21*(3*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 76*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 - 28*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 42*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 56*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 - 28*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 42*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 -
21*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 6)/((a^2 + 4*a^2*sin(d*x + c)/(cos(d*x + c) + 1) + 3*a^2*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 - 8*a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 14*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)
^4 + 14*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 8*a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3*a^2*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 - 4*a^2*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)
^10)*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.56 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {7 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {273 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2037 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 791 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 152}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{168 \, d} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/168*(7*(9*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) + 8)/(a^2*(tan(1/2*d*x + 1/2*c) - 1)^3) + (273*t
an(1/2*d*x + 1/2*c)^6 + 1155*tan(1/2*d*x + 1/2*c)^5 + 2450*tan(1/2*d*x + 1/2*c)^4 + 2870*tan(1/2*d*x + 1/2*c)^
3 + 2037*tan(1/2*d*x + 1/2*c)^2 + 791*tan(1/2*d*x + 1/2*c) + 152)/(a^2*(tan(1/2*d*x + 1/2*c) + 1)^7))/d

Mupad [B] (verification not implemented)

Time = 6.74 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.97 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+76\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-42\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-56\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+42\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+21\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )}{21\,a^2\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \]

[In]

int(1/(cos(c + d*x)^4*(a + a*sin(c + d*x))^2),x)

[Out]

(2*cos(c/2 + (d*x)/2)*(21*sin(c/2 + (d*x)/2)^9 - 6*cos(c/2 + (d*x)/2)^9 + 42*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x
)/2)^8 - 3*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 28*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^7 - 56*cos(c/2
+ (d*x)/2)^3*sin(c/2 + (d*x)/2)^6 - 42*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^5 + 28*cos(c/2 + (d*x)/2)^5*si
n(c/2 + (d*x)/2)^4 + 76*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^3 + 24*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)
^2))/(21*a^2*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))^3*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^7)