\(\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 40 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {2 a^4}{d (a-a \sin (c+d x))} \]

[Out]

a^3*ln(1-sin(d*x+c))/d+2*a^4/d/(a-a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45} \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 a^4}{d (a-a \sin (c+d x))}+\frac {a^3 \log (1-\sin (c+d x))}{d} \]

[In]

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

[Out]

(a^3*Log[1 - Sin[c + d*x]])/d + (2*a^4)/(d*(a - a*Sin[c + d*x]))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps \begin{align*} \text {integral}& = \frac {a^3 \text {Subst}\left (\int \frac {a+x}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^3 \text {Subst}\left (\int \left (\frac {2 a}{(a-x)^2}+\frac {1}{-a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {2 a^4}{d (a-a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sec ^2(c+d x) \left (\log (1-\sin (c+d x))+\frac {2}{1-\sin (c+d x)}\right ) (1-\sin (c+d x)) (1+\sin (c+d x))}{d} \]

[In]

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

[Out]

(a^3*Sec[c + d*x]^2*(Log[1 - Sin[c + d*x]] + 2/(1 - Sin[c + d*x]))*(1 - Sin[c + d*x])*(1 + Sin[c + d*x]))/d

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.80

method	result	size
risch	\(-i a^{3} x -\frac {2 i a^{3} c}{d}-\frac {4 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{\left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d}+\frac {2 a^{3} \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}\)	\(72\)
parallelrisch	\(-\frac {a^{3} \left (\left (\sin \left (d x +c \right )-1\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2-2 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 \sin \left (d x +c \right )\right )}{d \left (\sin \left (d x +c \right )-1\right )}\)	\(72\)
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(124\)
default	\(\frac {a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(124\)
norman	\(\frac {-\frac {8 a^{3}}{d}+\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {16 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {24 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a^{3} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(234\)

[In]

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

[Out]

-I*a^3*x-2*I*a^3/d*c-4*I*a^3*exp(I*(d*x+c))/(-I+exp(I*(d*x+c)))^2/d+2*a^3/d*ln(-I+exp(I*(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, a^{3} - {\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \]

[In]

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

[Out]

-(2*a^3 - (a^3*sin(d*x + c) - a^3)*log(-sin(d*x + c) + 1))/(d*sin(d*x + c) - d)

Sympy [F]

\[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \]

[In]

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

[Out]

(a^3*log(sin(d*x + c) - 1) - 2*a^3/(sin(d*x + c) - 1))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).

Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.30 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}}}{d} \]

[In]

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

[Out]

-(a^3*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 2*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + (3*a^3*tan(1/2*d*x + 1/2*c)
^2 - 10*a^3*tan(1/2*d*x + 1/2*c) + 3*a^3)/(tan(1/2*d*x + 1/2*c) - 1)^2)/d

Mupad [B] (verification not implemented)

Time = 6.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d}-\frac {2\,a^3}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]

[In]

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

[Out]

(a^3*log(sin(c + d*x) - 1))/d - (2*a^3)/(d*(sin(c + d*x) - 1))