3.3.14 \(\int \sec ^2(c+b x) \sin (a+b x) \, dx\) [214]

Optimal. Leaf size=34 \[ \frac {\cos (a-c) \sec (c+b x)}{b}+\frac {\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{b} \]

[Out]

cos(a-c)*sec(b*x+c)/b+arctanh(sin(b*x+c))*sin(a-c)/b

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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4676, 2686, 8, 3855} \begin {gather*} \frac {\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}+\frac {\cos (a-c) \sec (b x+c)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + b*x]^2*Sin[a + b*x],x]

[Out]

(Cos[a - c]*Sec[c + b*x])/b + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/b

Rule 8

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4676

Int[Sec[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Cos[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Sin[v - w],
Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rubi steps

\begin {align*} \int \sec ^2(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec (c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec (c+b x) \, dx\\ &=\frac {\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{b}+\frac {\cos (a-c) \text {Subst}(\int 1 \, dx,x,\sec (c+b x))}{b}\\ &=\frac {\cos (a-c) \sec (c+b x)}{b}+\frac {\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{b}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 88, normalized size = 2.59 \begin {gather*} \frac {\cos (a-c) \sec (c+b x)}{b}-\frac {2 i \text {ArcTan}\left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + b*x]^2*Sin[a + b*x],x]

[Out]

(Cos[a - c]*Sec[c + b*x])/b - ((2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/
(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(34)=68\).
time = 0.65, size = 346, normalized size = 10.18

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (b x +3 a \right )}+{\mathrm e}^{i \left (b x +a +2 c \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\)	\(115\)
default	\(\frac {\frac {4 \left (2 \sin \left (a \right ) \cos \left (c \right )-2 \cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+8 \cos \left (a \right ) \cos \left (c \right )+8 \sin \left (a \right ) \sin \left (c \right )}{\left (-4 \left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-4 \left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-4 \left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )\right ) \left (\cos \left (a \right ) \cos \left (c \right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin \left (a \right ) \sin \left (c \right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \cos \left (a \right ) \sin \left (c \right )-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )}+\frac {4 \left (2 \sin \left (a \right ) \cos \left (c \right )-2 \cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \sin \left (a \right ) \cos \left (c \right )+2 \cos \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}}\right )}{\left (-4 \left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-4 \left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-4 \left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-4 \left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )\right ) \sqrt {-\left (\cos ^{2}\left (c \right )\right ) \left (\sin ^{2}\left (a \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\cos ^{2}\left (c \right )\right )-\left (\sin ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )-\left (\cos ^{2}\left (a \right )\right ) \left (\sin ^{2}\left (c \right )\right )}}}{b}\)	\(346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(4*((2*sin(a)*cos(c)-2*cos(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-4*cos(a)^2*cos
(c)^2-4*cos(c)^2*sin(a)^2-4*cos(a)^2*sin(c)^2-4*sin(a)^2*sin(c)^2)/(cos(a)*cos(c)*tan(1/2*b*x+1/2*a)^2+sin(a)*
sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-2*tan(1/2*b*x+1/2*a)*sin(a)*cos(c)-cos(a)*cos(c
)-sin(a)*sin(c))+4*(2*sin(a)*cos(c)-2*cos(a)*sin(c))/(-4*cos(a)^2*cos(c)^2-4*cos(c)^2*sin(a)^2-4*cos(a)^2*sin(
c)^2-4*sin(a)^2*sin(c)^2)/(-cos(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)^2*sin(c)^2)^(1/2)*arc
tan(1/2*(2*(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)-2*sin(a)*cos(c)+2*cos(a)*sin(c))/(-cos(c)^2*sin(a)
^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)^2*sin(c)^2)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (34) = 68\).
time = 0.55, size = 387, normalized size = 11.38 \begin {gather*} \frac {2 \, {\left (\cos \left (b x + 2 \, a\right ) + \cos \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) + 2 \, \cos \left (b x + 2 \, a\right ) \cos \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \cos \left (a\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \sin \left (-a + c\right ) + \sin \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) \sin \left (-a + c\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \sin \left (-a + c\right )\right )} \log \left (\frac {\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) + 2 \, {\left (\sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) + 2 \, \sin \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \sin \left (b x + 2 \, c\right ) \sin \left (a\right )}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 2*cos(b*x + 2*a)*cos(a) + 2*cos(b*x + 2*c)*cos
(a) + (cos(2*b*x + a + 2*c)^2*sin(-a + c) + 2*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(2*b*x + a + 2*c)^2
*sin(-a + c) + 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log((cos(b*x + 2
*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)/(cos(b*x +
2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)) + 2*(si
n(b*x + 2*a) + sin(b*x + 2*c))*sin(2*b*x + a + 2*c) + 2*sin(b*x + 2*a)*sin(a) + 2*sin(b*x + 2*c)*sin(a))/(b*co
s(2*b*x + a + 2*c)^2 + 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*x + a + 2*c)^2 + 2*b*sin(2*b*x + a + 2*c)*s
in(a) + (cos(a)^2 + sin(a)^2)*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
time = 2.92, size = 69, normalized size = 2.03 \begin {gather*} -\frac {\cos \left (b x + c\right ) \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - \cos \left (b x + c\right ) \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 2 \, \cos \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(cos(b*x + c)*log(sin(b*x + c) + 1)*sin(-a + c) - cos(b*x + c)*log(-sin(b*x + c) + 1)*sin(-a + c) - 2*cos
(-a + c))/(b*cos(b*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (27) = 54\).
time = 91.13, size = 5545, normalized size = 163.09 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(tan(b*x/2))/b, Eq(c, -pi/2) | Eq(c, pi/2)), (0, Eq(b, 0)), (-2*log(tan(b*x/2) - tan(c/2)/(tan(c
/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(
c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) -
1) - 1/(tan(c/2) - 1))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) -
4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 8*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1
))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(
c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/
2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(
b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)/(b*tan
(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**
2 + b) + 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)*
*4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b)
- 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2)**2 - b
*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 8*log(tan(b*x/2)
+ tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)
**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + tan(c/2
)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b
*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + tan(c/2)/(tan(c/
2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2)
- 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*tan(c/2)**4/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4
- 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 4*tan(c/2)**3*tan(b*x/2)/(b*t
an(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)
**2 + b) - 4*tan(c/2)*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4
*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)*
*3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b), True))*cos(a) + Piecewise((-1/(b*sin(b*x)), Eq
(c, -pi/2) | Eq(c, pi/2)), (x/cos(c)**2, Eq(b, 0)), (-log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) -
1))*tan(c/2)**6*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*t
an(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/
2) + b*tan(b*x/2)**2 - b) + log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**6/(b*tan(c/
2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4
- b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 4*log(tan(b*
x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**5*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan
(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)
**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan(b*x/2) - tan(c/2)/(tan(c/2) -
1) - 1/(tan(c/2) - 1))*tan(c/2)**4*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)*
*5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*
b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 3*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*
tan(c/2)**4/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/
2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**
2 - b) - 8*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**3*tan(b*x/2)/(b*tan(c/2)**6*
tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*t
an(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 3*log(tan(b*x/2) -
tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**2*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/
2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2
+ b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1)
- 1/(tan(c/2) - 1))*tan(c/2)**2/(b*tan(c/2)**6...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (34) = 68\).
time = 0.44, size = 248, normalized size = 7.29 \begin {gather*} \frac {2 \, {\left (\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{2} - 1\right )}}\right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c)
+ 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*ta
n(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c) - 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*
a)^2 + tan(1/2*c)^2 + 1) - (tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2
+ 1)/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*(tan(1/2*b*x + 1/2*c)^2 - 1)))/b

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Mupad [B]
time = 5.35, size = 254, normalized size = 7.47 \begin {gather*} \frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/cos(c + b*x)^2,x)

[Out]

(exp(a*1i + b*x*1i)*(exp(a*2i - c*2i) + 1))/(b*(exp(a*2i - c*2i) + exp(a*2i + b*x*2i))) + (log(exp(a*1i)*exp(b
*x*1i)*(exp(a*2i)*exp(-c*2i)*1i - 1i) - (exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-
c*2i))^(1/2))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2)) - (log(exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*
exp(-c*2i)*1i - 1i) + (exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-c*2i))^(1/2))*(exp
(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2))

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