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

Optimal. Leaf size=87 \[ \frac{2 \sin ^3(c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2} \]

[Out]

(7*x)/(8*a^2) - (2*Sin[c + d*x])/(a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d
*x])/(4*a^2*d) + (2*Sin[c + d*x]^3)/(3*a^2*d)

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Rubi [A]  time = 0.232528, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3872, 2869, 2757, 2635, 8, 2633} \[ \frac{2 \sin ^3(c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*x)/(8*a^2) - (2*Sin[c + d*x])/(a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d
*x])/(4*a^2*d) + (2*Sin[c + d*x]^3)/(3*a^2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2869

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[a^(2*m), Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f,
n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]

Rule 2757

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

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

Rubi steps

\begin{align*} \int \frac{\sin ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\int \cos ^2(c+d x) (-a+a \cos (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cos ^2(c+d x)-2 a^2 \cos ^3(c+d x)+a^2 \cos ^4(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cos ^2(c+d x) \, dx}{a^2}+\frac{\int \cos ^4(c+d x) \, dx}{a^2}-\frac{2 \int \cos ^3(c+d x) \, dx}{a^2}\\ &=\frac{\cos (c+d x) \sin (c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{\int 1 \, dx}{2 a^2}+\frac{3 \int \cos ^2(c+d x) \, dx}{4 a^2}+\frac{2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=\frac{x}{2 a^2}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{2 \sin ^3(c+d x)}{3 a^2 d}+\frac{3 \int 1 \, dx}{8 a^2}\\ &=\frac{7 x}{8 a^2}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{2 \sin ^3(c+d x)}{3 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.544488, size = 91, normalized size = 1.05 \[ \frac{\cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (-144 \sin (c+d x)+48 \sin (2 (c+d x))-16 \sin (3 (c+d x))+3 \sin (4 (c+d x))+2 \tan \left (\frac{c}{2}\right )+84 d x\right )}{24 a^2 d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(84*d*x - 144*Sin[c + d*x] + 48*Sin[2*(c + d*x)] - 16*Sin[3*(c + d*x)] + 3*
Sin[4*(c + d*x)] + 2*Tan[c/2]))/(24*a^2*d*(1 + Sec[c + d*x])^2)

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Maple [A]  time = 0.089, size = 154, normalized size = 1.8 \begin{align*} -{\frac{25}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{83}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{77}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{7}{4\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{7}{4\,d{a}^{2}}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7-83/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x
+1/2*c)^5-77/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3-7/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan
(1/2*d*x+1/2*c)+7/4/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.52328, size = 278, normalized size = 3.2 \begin{align*} -\frac{\frac{\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{83 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{21 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 83*sin(d*x + c)^5/(cos(d
*x + c) + 1)^5 + 75*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^2 + 4*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*
a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^2*sin(d*x + c)^8/(cos(
d*x + c) + 1)^8) - 21*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.72595, size = 135, normalized size = 1.55 \begin{align*} \frac{21 \, d x +{\left (6 \, \cos \left (d x + c\right )^{3} - 16 \, \cos \left (d x + c\right )^{2} + 21 \, \cos \left (d x + c\right ) - 32\right )} \sin \left (d x + c\right )}{24 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(21*d*x + (6*cos(d*x + c)^3 - 16*cos(d*x + c)^2 + 21*cos(d*x + c) - 32)*sin(d*x + c))/(a^2*d)

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

[Out]

Timed out

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Giac [A]  time = 1.36183, size = 117, normalized size = 1.34 \begin{align*} \frac{\frac{21 \,{\left (d x + c\right )}}{a^{2}} - \frac{2 \,{\left (75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 83 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 77 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(21*(d*x + c)/a^2 - 2*(75*tan(1/2*d*x + 1/2*c)^7 + 83*tan(1/2*d*x + 1/2*c)^5 + 77*tan(1/2*d*x + 1/2*c)^3
+ 21*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^2))/d