3.1.12 \(\int \frac {\sin ^2(a+b x)}{c+d x} \, dx\) [12]

Optimal. Leaf size=78 \[ -\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]

[Out]

-1/2*Ci(2*b*c/d+2*b*x)*cos(2*a-2*b*c/d)/d+1/2*ln(d*x+c)/d+1/2*Si(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3384, 3380, 3383} \begin {gather*} -\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^2/(c + d*x),x]

[Out]

-1/2*(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d + Log[c + d*x]/(2*d) + (Sin[2*a - (2*b*c)/d]*SinI
ntegral[(2*b*c)/d + 2*b*x])/(2*d)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin {align*} \int \frac {\sin ^2(a+b x)}{c+d x} \, dx &=\int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx\\ &=\frac {\log (c+d x)}{2 d}-\frac {1}{2} \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx\\ &=\frac {\log (c+d x)}{2 d}-\frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 65, normalized size = 0.83 \begin {gather*} \frac {-\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b (c+d x)}{d}\right )+\log (c+d x)+\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^2/(c + d*x),x]

[Out]

(-(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d]) + Log[c + d*x] + Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b
*(c + d*x))/d])/(2*d)

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 114, normalized size = 1.46

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {2 i \left (d a -c b \right )}{d}} \expIntegral \left (1, 2 i b x +2 i a -\frac {2 i \left (d a -c b \right )}{d}\right )}{4 d}+\frac {{\mathrm e}^{\frac {2 i \left (d a -c b \right )}{d}} \expIntegral \left (1, -2 i b x -2 i a -\frac {2 \left (-i a d +i b c \right )}{d}\right )}{4 d}+\frac {\ln \left (d x +c \right )}{2 d}\)	\(107\)
derivativedivides	\(\frac {\frac {b \ln \left (-d a +c b +d \left (b x +a \right )\right )}{2 d}-\frac {b \left (-\frac {2 \sinIntegral \left (-2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{4}}{b}\)	\(114\)
default	\(\frac {\frac {b \ln \left (-d a +c b +d \left (b x +a \right )\right )}{2 d}-\frac {b \left (-\frac {2 \sinIntegral \left (-2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{4}}{b}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/2*b*ln(-d*a+c*b+d*(b*x+a))/d-1/4*b*(-2*Si(-2*b*x-2*a-2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d+2*Ci(2*b*x+2
*a+2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d))

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.35, size = 162, normalized size = 2.08 \begin {gather*} \frac {b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, b \log \left (b c + {\left (b x + a\right )} d - a d\right )}{4 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*(exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -2*(-I*b*c - I*(b*x + a)*d
+ I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b*(I*exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*exp_int
egral_e(1, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d) + 2*b*log(b*c + (b*x + a)*d - a*d))/(
b*d)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 88, normalized size = 1.13 \begin {gather*} -\frac {{\left (\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, \log \left (d x + c\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((cos_integral(2*(b*d*x + b*c)/d) + cos_integral(-2*(b*d*x + b*c)/d))*cos(-2*(b*c - a*d)/d) - 2*sin(-2*(b
*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) - 2*log(d*x + c))/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x)**2/(c + d*x), x)

________________________________________________________________________________________

Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.24, size = 612, normalized size = 7.85 \begin {gather*} \frac {2 \, \log \left ({\left | d x + c \right |}\right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) + 4 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} - 4 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \log \left ({\left | d x + c \right |}\right ) \tan \left (a\right )^{2} + \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} + \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} - 4 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) - 4 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) + 2 \, \log \left ({\left | d x + c \right |}\right ) \tan \left (\frac {b c}{d}\right )^{2} + \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} + \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) + 4 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) + 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) - 4 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (\frac {b c}{d}\right ) + 2 \, \log \left ({\left | d x + c \right |}\right ) - \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) - \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right )}{4 \, {\left (d \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + d \tan \left (a\right )^{2} + d \tan \left (\frac {b c}{d}\right )^{2} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*log(abs(d*x + c))*tan(a)^2*tan(b*c/d)^2 - real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^
2 - real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d
))*tan(a)^2*tan(b*c/d) - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) + 4*sin_integral(2*(b
*d*x + b*c)/d)*tan(a)^2*tan(b*c/d) - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 2*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d)^
2 + 2*log(abs(d*x + c))*tan(a)^2 + real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 + real_part(cos_integral(
-2*b*x - 2*b*c/d))*tan(a)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) - 4*real_part(cos_i
ntegral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 2*log(abs(d*x + c))*tan(b*c/d)^2 + real_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*c/d)^2 + real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 2*imag_part(cos_integral(
2*b*x + 2*b*c/d))*tan(a) - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) + 4*sin_integral(2*(b*d*x + b*c)
/d)*tan(a) - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d
))*tan(b*c/d) - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) + 2*log(abs(d*x + c)) - real_part(cos_integral(2*
b*x + 2*b*c/d)) - real_part(cos_integral(-2*b*x - 2*b*c/d)))/(d*tan(a)^2*tan(b*c/d)^2 + d*tan(a)^2 + d*tan(b*c
/d)^2 + d)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (a+b\,x\right )}^2}{c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^2/(c + d*x),x)

[Out]

int(sin(a + b*x)^2/(c + d*x), x)

________________________________________________________________________________________