3.930 \(\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=24 \[ x (b B-a C)+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

(b*B - a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d

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Rubi [A]  time = 0.0232022, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {24, 3770} \[ x (b B-a C)+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]

[Out]

(b*B - a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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 \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\int \left (b^2 (b B-a C)+b^3 C \sec (c+d x)\right ) \, dx}{b^2}\\ &=(b B-a C) x+(b C) \int \sec (c+d x) \, dx\\ &=(b B-a C) x+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.0110044, size = 23, normalized size = 0.96 \[ -a C x+b B x+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]

[Out]

b*B*x - a*C*x + (b*C*ArcTanh[Sin[c + d*x]])/d

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Maple [A]  time = 0.046, size = 46, normalized size = 1.9 \begin{align*} Bbx-aCx+{\frac{Bbc}{d}}+{\frac{Cb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{Cac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*a*b-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x)

[Out]

B*b*x-a*C*x+1/d*B*b*c+1/d*C*b*ln(sec(d*x+c)+tan(d*x+c))-1/d*C*a*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.49816, size = 115, normalized size = 4.79 \begin{align*} -\frac{2 \,{\left (C a - B b\right )} d x - C b \log \left (\sin \left (d x + c\right ) + 1\right ) + C b \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*(C*a - B*b)*d*x - C*b*log(sin(d*x + c) + 1) + C*b*log(-sin(d*x + c) + 1))/d

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Sympy [A]  time = 3.51186, size = 75, normalized size = 3.12 \begin{align*} \begin{cases} - \frac{- B b \left (c + d x\right ) + C a \left (c + d x\right ) - C b \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (B a b + B b^{2} \sec{\left (c \right )} - C a^{2} + C b^{2} \sec ^{2}{\left (c \right )}\right )}{a + b \sec{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a**2*C+b**2*B*sec(d*x+c)+b**2*C*sec(d*x+c)**2)/(a+b*sec(d*x+c)),x)

[Out]

Piecewise((-(-B*b*(c + d*x) + C*a*(c + d*x) - C*b*log(tan(c + d*x) + sec(c + d*x)))/d, Ne(d, 0)), (x*(B*a*b +
B*b**2*sec(c) - C*a**2 + C*b**2*sec(c)**2)/(a + b*sec(c)), True))

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Giac [B]  time = 1.25493, size = 72, normalized size = 3. \begin{align*} \frac{C b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - C b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) -{\left (C a - B b\right )}{\left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

(C*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - C*b*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - (C*a - B*b)*(d*x + c))/d