3.1.24 \(\int \frac {1}{(a+b x^2) (c+d x^2)} \, dx\) [24]

Optimal. Leaf size=70 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)} \]

[Out]

arctan(x*b^(1/2)/a^(1/2))*b^(1/2)/(-a*d+b*c)/a^(1/2)-arctan(x*d^(1/2)/c^(1/2))*d^(1/2)/(-a*d+b*c)/c^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {400, 211} \begin {gather*} \frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*(c + d*x^2)),x]

[Out]

(Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) - (Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(
b*c - a*d))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 400

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {b \int \frac {1}{a+b x^2} \, dx}{b c-a d}-\frac {d \int \frac {1}{c+d x^2} \, dx}{b c-a d}\\ &=\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.87 \begin {gather*} \frac {\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c}}}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)*(c + d*x^2)),x]

[Out]

((Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[c])/(b*c - a*d)

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Maple [A]
time = 0.11, size = 55, normalized size = 0.79

method	result	size
default	\(-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}}+\frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}}\)	\(55\)
risch	\(\frac {\sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 a \left (a d -b c \right )}-\frac {\sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{2 a \left (a d -b c \right )}+\frac {\sqrt {-c d}\, \ln \left (d x +\sqrt {-c d}\right )}{2 c \left (a d -b c \right )}-\frac {\sqrt {-c d}\, \ln \left (d x -\sqrt {-c d}\right )}{2 c \left (a d -b c \right )}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

-b/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+d/(a*d-b*c)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2))

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Maxima [A]
time = 0.51, size = 54, normalized size = 0.77 \begin {gather*} \frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) - d*arctan(d*x/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))

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Fricas [A]
time = 1.70, size = 292, normalized size = 4.17 \begin {gather*} \left [-\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, \frac {\sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{b c - a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) -
c)/(d*x^2 + c)))/(b*c - a*d), -1/2*(2*sqrt(d/c)*arctan(x*sqrt(d/c)) + sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a
) - a)/(b*x^2 + a)))/(b*c - a*d), 1/2*(2*sqrt(b/a)*arctan(x*sqrt(b/a)) - sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d
/c) - c)/(d*x^2 + c)))/(b*c - a*d), (sqrt(b/a)*arctan(x*sqrt(b/a)) - sqrt(d/c)*arctan(x*sqrt(d/c)))/(b*c - a*d
)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (60) = 120\).
time = 2.73, size = 712, normalized size = 10.17 \begin {gather*} \frac {\sqrt {- \frac {b}{a}} \log {\left (x + \frac {- \frac {a^{4} c d^{3} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b c^{2} d^{2} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{2} c^{3} d \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} d^{2} \sqrt {- \frac {b}{a}}}{a d - b c} - \frac {a b^{3} c^{4} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{2} c^{2} \sqrt {- \frac {b}{a}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {b}{a}} \log {\left (x + \frac {\frac {a^{4} c d^{3} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b c^{2} d^{2} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{2} c^{3} d \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} d^{2} \sqrt {- \frac {b}{a}}}{a d - b c} + \frac {a b^{3} c^{4} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{2} c^{2} \sqrt {- \frac {b}{a}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {d}{c}} \log {\left (x + \frac {- \frac {a^{4} c d^{3} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b c^{2} d^{2} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{2} c^{3} d \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} d^{2} \sqrt {- \frac {d}{c}}}{a d - b c} - \frac {a b^{3} c^{4} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{2} c^{2} \sqrt {- \frac {d}{c}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {d}{c}} \log {\left (x + \frac {\frac {a^{4} c d^{3} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b c^{2} d^{2} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{2} c^{3} d \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} d^{2} \sqrt {- \frac {d}{c}}}{a d - b c} + \frac {a b^{3} c^{4} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{2} c^{2} \sqrt {- \frac {d}{c}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)/(d*x**2+c),x)

[Out]

sqrt(-b/a)*log(x + (-a**4*c*d**3*(-b/a)**(3/2)/(a*d - b*c)**3 + a**3*b*c**2*d**2*(-b/a)**(3/2)/(a*d - b*c)**3
+ a**2*b**2*c**3*d*(-b/a)**(3/2)/(a*d - b*c)**3 - a**2*d**2*sqrt(-b/a)/(a*d - b*c) - a*b**3*c**4*(-b/a)**(3/2)
/(a*d - b*c)**3 - b**2*c**2*sqrt(-b/a)/(a*d - b*c))/(b*d))/(2*(a*d - b*c)) - sqrt(-b/a)*log(x + (a**4*c*d**3*(
-b/a)**(3/2)/(a*d - b*c)**3 - a**3*b*c**2*d**2*(-b/a)**(3/2)/(a*d - b*c)**3 - a**2*b**2*c**3*d*(-b/a)**(3/2)/(
a*d - b*c)**3 + a**2*d**2*sqrt(-b/a)/(a*d - b*c) + a*b**3*c**4*(-b/a)**(3/2)/(a*d - b*c)**3 + b**2*c**2*sqrt(-
b/a)/(a*d - b*c))/(b*d))/(2*(a*d - b*c)) + sqrt(-d/c)*log(x + (-a**4*c*d**3*(-d/c)**(3/2)/(a*d - b*c)**3 + a**
3*b*c**2*d**2*(-d/c)**(3/2)/(a*d - b*c)**3 + a**2*b**2*c**3*d*(-d/c)**(3/2)/(a*d - b*c)**3 - a**2*d**2*sqrt(-d
/c)/(a*d - b*c) - a*b**3*c**4*(-d/c)**(3/2)/(a*d - b*c)**3 - b**2*c**2*sqrt(-d/c)/(a*d - b*c))/(b*d))/(2*(a*d
- b*c)) - sqrt(-d/c)*log(x + (a**4*c*d**3*(-d/c)**(3/2)/(a*d - b*c)**3 - a**3*b*c**2*d**2*(-d/c)**(3/2)/(a*d -
b*c)**3 - a**2*b**2*c**3*d*(-d/c)**(3/2)/(a*d - b*c)**3 + a**2*d**2*sqrt(-d/c)/(a*d - b*c) + a*b**3*c**4*(-d/
c)**(3/2)/(a*d - b*c)**3 + b**2*c**2*sqrt(-d/c)/(a*d - b*c))/(b*d))/(2*(a*d - b*c))

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Giac [A]
time = 1.84, size = 54, normalized size = 0.77 \begin {gather*} \frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) - d*arctan(d*x/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))

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Mupad [B]
time = 0.32, size = 135, normalized size = 1.93 \begin {gather*} \frac {\ln \left (b\,x-\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,a^2\,d-2\,a\,b\,c}-\frac {\ln \left (d\,x+\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,\left (b\,c^2-a\,c\,d\right )}-\frac {\ln \left (b\,x+\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,\left (a^2\,d-a\,b\,c\right )}+\frac {\ln \left (d\,x-\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,b\,c^2-2\,a\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)*(c + d*x^2)),x)

[Out]

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

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