3.250 \(\int (c+d x) \sec (a+b x) \tan (a+b x) \, dx\)
Optimal. Leaf size=29 \[ \frac{(c+d x) \sec (a+b x)}{b}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2} \]
[Out]
-((d*ArcTanh[Sin[a + b*x]])/b^2) + ((c + d*x)*Sec[a + b*x])/b
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Rubi [A] time = 0.0193704, antiderivative size = 29, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{4409, 3770} \[ \frac{(c+d x) \sec (a+b x)}{b}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)*Sec[a + b*x]*Tan[a + b*x],x]
[Out]
-((d*ArcTanh[Sin[a + b*x]])/b^2) + ((c + d*x)*Sec[a + b*x])/b
Rule 4409
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
((c + d*x)^m*Sec[a + b*x]^n)/(b*n), x] - Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int (c+d x) \sec (a+b x) \tan (a+b x) \, dx &=\frac{(c+d x) \sec (a+b x)}{b}-\frac{d \int \sec (a+b x) \, dx}{b}\\ &=-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{(c+d x) \sec (a+b x)}{b}\\ \end{align*}
Mathematica [B] time = 0.0471758, size = 93, normalized size = 3.21 \[ \frac{d \log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )-\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b^2}-\frac{d \log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )+\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b^2}+\frac{c \sec (a+b x)}{b}+\frac{d x \sec (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)*Sec[a + b*x]*Tan[a + b*x],x]
[Out]
(d*Log[Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2]])/b^2 - (d*Log[Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]])/b^2 +
(c*Sec[a + b*x])/b + (d*x*Sec[a + b*x])/b
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Maple [A] time = 0.022, size = 49, normalized size = 1.7 \begin{align*}{\frac{dx}{b\cos \left ( bx+a \right ) }}-{\frac{d\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{{b}^{2}}}+{\frac{c}{b\cos \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)*sec(b*x+a)*tan(b*x+a),x)
[Out]
1/b*d/cos(b*x+a)*x-1/b^2*d*ln(sec(b*x+a)+tan(b*x+a))+1/b*c/cos(b*x+a)
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Maxima [B] time = 1.49983, size = 350, normalized size = 12.07 \begin{align*} \frac{\frac{{\left (4 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) + 4 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 4 \,{\left (b x + a\right )} \cos \left (b x + a\right ) -{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \sin \left (b x + a\right ) + 1\right ) +{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \sin \left (b x + a\right ) + 1\right )\right )} d}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b} + \frac{2 \, c}{\cos \left (b x + a\right )} - \frac{2 \, a d}{b \cos \left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*sec(b*x+a)*tan(b*x+a),x, algorithm="maxima")
[Out]
1/2*((4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + a)*sin(2*b*x + 2*a)*sin(b*x + a) + 4*(b*x + a)*cos(
b*x + a) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a
)^2 + 2*sin(b*x + a) + 1) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a
)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1))*d/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) +
1)*b) + 2*c/cos(b*x + a) - 2*a*d/(b*cos(b*x + a)))/b
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Fricas [B] time = 0.492755, size = 163, normalized size = 5.62 \begin{align*} \frac{2 \, b d x - d \cos \left (b x + a\right ) \log \left (\sin \left (b x + a\right ) + 1\right ) + d \cos \left (b x + a\right ) \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, b c}{2 \, b^{2} \cos \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*sec(b*x+a)*tan(b*x+a),x, algorithm="fricas")
[Out]
1/2*(2*b*d*x - d*cos(b*x + a)*log(sin(b*x + a) + 1) + d*cos(b*x + a)*log(-sin(b*x + a) + 1) + 2*b*c)/(b^2*cos(
b*x + a))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) \tan{\left (a + b x \right )} \sec{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*sec(b*x+a)*tan(b*x+a),x)
[Out]
Integral((c + d*x)*tan(a + b*x)*sec(a + b*x), x)
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Giac [B] time = 1.68498, size = 2075, normalized size = 71.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*sec(b*x+a)*tan(b*x+a),x, algorithm="giac")
[Out]
1/2*(2*b*d*x*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*b*c*tan(1/2*b*x)^2*tan(1/2*a)^2 + d*log(2*(tan(1/2*a)^2 + 1)/(tan
(1/2*b*x)^4*tan(1/2*a)^2 + 2*tan(1/2*b*x)^4*tan(1/2*a) + 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*ta
n(1/2*b*x)^2*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3 + 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 -
2*tan(1/2*b*x) - 2*tan(1/2*a) + 1))*tan(1/2*b*x)^2*tan(1/2*a)^2 - d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*
tan(1/2*a)^2 - 2*tan(1/2*b*x)^4*tan(1/2*a) - 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2
*tan(1/2*a)^2 + 2*tan(1/2*b*x)^3 - 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 + 2*tan(1/2*b
*x) + 2*tan(1/2*a) + 1))*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*b*d*x*tan(1/2*b*x)^2 + 2*b*d*x*tan(1/2*a)^2 + 2*b*c*t
an(1/2*b*x)^2 - d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 + 2*tan(1/2*b*x)^4*tan(1/2*a) + 2*tan(
1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3 + 2*tan(1/2*b*x)*t
an(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 - 2*tan(1/2*b*x) - 2*tan(1/2*a) + 1))*tan(1/2*b*x)^2 + d*log(2*(
tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^4*tan(1/2*a) - 2*tan(1/2*b*x)^3*tan(1/2*a)^2 +
tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*tan(1/2*b*x)^3 - 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b
*x)^2 + tan(1/2*a)^2 + 2*tan(1/2*b*x) + 2*tan(1/2*a) + 1))*tan(1/2*b*x)^2 - 4*d*log(2*(tan(1/2*a)^2 + 1)/(tan(
1/2*b*x)^4*tan(1/2*a)^2 + 2*tan(1/2*b*x)^4*tan(1/2*a) + 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan
(1/2*b*x)^2*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3 + 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 -
2*tan(1/2*b*x) - 2*tan(1/2*a) + 1))*tan(1/2*b*x)*tan(1/2*a) + 4*d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan
(1/2*a)^2 - 2*tan(1/2*b*x)^4*tan(1/2*a) - 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*ta
n(1/2*a)^2 + 2*tan(1/2*b*x)^3 - 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 + 2*tan(1/2*b*x)
+ 2*tan(1/2*a) + 1))*tan(1/2*b*x)*tan(1/2*a) + 2*b*c*tan(1/2*a)^2 - d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^
4*tan(1/2*a)^2 + 2*tan(1/2*b*x)^4*tan(1/2*a) + 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)
^2*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3 + 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 - 2*tan(1/2
*b*x) - 2*tan(1/2*a) + 1))*tan(1/2*a)^2 + d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*
b*x)^4*tan(1/2*a) - 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*tan(1/2
*b*x)^3 - 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 + 2*tan(1/2*b*x) + 2*tan(1/2*a) + 1))*
tan(1/2*a)^2 + 2*b*d*x + 2*b*c + d*log(2*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 + 2*tan(1/2*b*x)^4*ta
n(1/2*a) + 2*tan(1/2*b*x)^3*tan(1/2*a)^2 + tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3 +
2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2 + tan(1/2*a)^2 - 2*tan(1/2*b*x) - 2*tan(1/2*a) + 1)) - d*log(2
*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^4*tan(1/2*a) - 2*tan(1/2*b*x)^3*tan(1/2*a)^2
+ tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*tan(1/2*b*x)^3 - 2*tan(1/2*b*x)*tan(1/2*a)^2 + 2*tan(1/2
*b*x)^2 + tan(1/2*a)^2 + 2*tan(1/2*b*x) + 2*tan(1/2*a) + 1)))/(b^2*tan(1/2*b*x)^2*tan(1/2*a)^2 - b^2*tan(1/2*b
*x)^2 - 4*b^2*tan(1/2*b*x)*tan(1/2*a) - b^2*tan(1/2*a)^2 + b^2)