\(\int \frac {1}{(3+5 \sin (c+d x))^2} \, dx\) [35]

Optimal result

Integrand size = 12, antiderivative size = 88 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=\frac {3 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))} \]

[Out]

3/64*ln(3*cos(1/2*d*x+1/2*c)+sin(1/2*d*x+1/2*c))/d-3/64*ln(cos(1/2*d*x+1/2*c)+3*sin(1/2*d*x+1/2*c))/d-5/16*cos
(d*x+c)/d/(3+5*sin(d*x+c))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 12, 2739, 630, 31} \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=-\frac {5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \]

[In]

Int[(3 + 5*Sin[c + d*x])^(-2),x]

[Out]

(3*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(64*d) - (3*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]])/(64*d)
- (5*Cos[c + d*x])/(16*d*(3 + 5*Sin[c + d*x]))

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}+\frac {1}{16} \int -\frac {3}{3+5 \sin (c+d x)} \, dx \\ & = -\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}-\frac {3}{16} \int \frac {1}{3+5 \sin (c+d x)} \, dx \\ & = -\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}-\frac {3 \text {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \\ & = -\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}-\frac {9 \text {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {9 \text {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \\ & = \frac {3 \log \left (3+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (1+3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=\frac {9 \left (\log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {3}{\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )}\right )}{192 d} \]

[In]

Integrate[(3 + 5*Sin[c + d*x])^(-2),x]

[Out]

(9*(Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]]) + 20*Sin[(c + d*x
)/2]*((3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-1) + 3/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(192*d)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{64}-\frac {5}{48 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{64}}{d}\)	\(68\)
default	\(\frac {-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{64}-\frac {5}{48 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{64}}{d}\)	\(68\)
risch	\(-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}+5 i}{8 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-5+6 i {\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {4}{5}+\frac {3 i}{5}\right )}{64 d}-\frac {3 \ln \left (-\frac {4}{5}+\frac {3 i}{5}+{\mathrm e}^{i \left (d x +c \right )}\right )}{64 d}\)	\(86\)
norman	\(\frac {-\frac {5}{8 d}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d}}{3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{64 d}-\frac {3 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{64 d}\)	\(87\)
parallelrisch	\(\frac {45 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right ) \sin \left (d x +c \right )-45 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{3}\right ) \sin \left (d x +c \right )-100 \sin \left (d x +c \right )-60 \cos \left (d x +c \right )+27 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )-27 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{3}\right )-60}{192 d \left (3+5 \sin \left (d x +c \right )\right )}\)	\(104\)

[In]

int(1/(3+5*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-5/16/(tan(1/2*d*x+1/2*c)+3)+3/64*ln(tan(1/2*d*x+1/2*c)+3)-5/48/(3*tan(1/2*d*x+1/2*c)+1)-3/64*ln(3*tan(1/
2*d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=\frac {3 \, {\left (5 \, \sin \left (d x + c\right ) + 3\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 3 \, {\left (5 \, \sin \left (d x + c\right ) + 3\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 40 \, \cos \left (d x + c\right )}{128 \, {\left (5 \, d \sin \left (d x + c\right ) + 3 \, d\right )}} \]

[In]

integrate(1/(3+5*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/128*(3*(5*sin(d*x + c) + 3)*log(4*cos(d*x + c) + 3*sin(d*x + c) + 5) - 3*(5*sin(d*x + c) + 3)*log(-4*cos(d*x
+ c) + 3*sin(d*x + c) + 5) - 40*cos(d*x + c))/(5*d*sin(d*x + c) + 3*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (78) = 156\).

Time = 0.72 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.30 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=\begin {cases} \frac {x}{\left (3 - 5 \sin {\left (2 \operatorname {atan}{\left (\frac {1}{3} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {1}{3} \right )} \\\frac {x}{\left (3 - 5 \sin {\left (2 \operatorname {atan}{\left (3 \right )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (3 \right )} \\\frac {x}{\left (5 \sin {\left (c \right )} + 3\right )^{2}} & \text {for}\: d = 0 \\\frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} + \frac {90 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} + \frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {27 \log {\left (3 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {90 \log {\left (3 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {27 \log {\left (3 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {200 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {120}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(3+5*sin(d*x+c))**2,x)

[Out]

Piecewise((x/(3 - 5*sin(2*atan(1/3)))**2, Eq(c, -d*x - 2*atan(1/3))), (x/(3 - 5*sin(2*atan(3)))**2, Eq(c, -d*x
- 2*atan(3))), (x/(5*sin(c) + 3)**2, Eq(d, 0)), (27*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**2/(576*d*tan(
c/2 + d*x/2)**2 + 1920*d*tan(c/2 + d*x/2) + 576*d) + 90*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)/(576*d*tan(
c/2 + d*x/2)**2 + 1920*d*tan(c/2 + d*x/2) + 576*d) + 27*log(tan(c/2 + d*x/2) + 3)/(576*d*tan(c/2 + d*x/2)**2 +
1920*d*tan(c/2 + d*x/2) + 576*d) - 27*log(3*tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)**2/(576*d*tan(c/2 + d*x/2)
**2 + 1920*d*tan(c/2 + d*x/2) + 576*d) - 90*log(3*tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)/(576*d*tan(c/2 + d*x/
2)**2 + 1920*d*tan(c/2 + d*x/2) + 576*d) - 27*log(3*tan(c/2 + d*x/2) + 1)/(576*d*tan(c/2 + d*x/2)**2 + 1920*d*
tan(c/2 + d*x/2) + 576*d) - 200*tan(c/2 + d*x/2)/(576*d*tan(c/2 + d*x/2)**2 + 1920*d*tan(c/2 + d*x/2) + 576*d)
- 120/(576*d*tan(c/2 + d*x/2)**2 + 1920*d*tan(c/2 + d*x/2) + 576*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=-\frac {\frac {40 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}}{\frac {10 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 3} + 9 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{192 \, d} \]

[In]

integrate(1/(3+5*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/192*(40*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 3)/(10*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos
(d*x + c) + 1)^2 + 3) + 9*log(3*sin(d*x + c)/(cos(d*x + c) + 1) + 1) - 9*log(sin(d*x + c)/(cos(d*x + c) + 1) +
3))/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=-\frac {\frac {40 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}}{3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3} + 9 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{192 \, d} \]

[In]

integrate(1/(3+5*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/192*(40*(5*tan(1/2*d*x + 1/2*c) + 3)/(3*tan(1/2*d*x + 1/2*c)^2 + 10*tan(1/2*d*x + 1/2*c) + 3) + 9*log(abs(3
*tan(1/2*d*x + 1/2*c) + 1)) - 9*log(abs(tan(1/2*d*x + 1/2*c) + 3)))/d

Mupad [B] (verification not implemented)

Time = 5.72 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(3+5 \sin (c+d x))^2} \, dx=\frac {3\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{32\,d}-\frac {\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{72}+\frac {5}{24}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1\right )} \]

[In]

int(1/(5*sin(c + d*x) + 3)^2,x)

[Out]

(3*atanh((3*tan(c/2 + (d*x)/2))/4 + 5/4))/(32*d) - ((25*tan(c/2 + (d*x)/2))/72 + 5/24)/(d*((10*tan(c/2 + (d*x)
/2))/3 + tan(c/2 + (d*x)/2)^2 + 1))