Optimal. Leaf size=66 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\tan (c+d x)}{2 a d \left (a+b \tan ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3756, 205, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\tan (c+d x)}{2 a d \left (a+b \tan ^2(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 3756
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan (c+d x)}{2 a d \left (a+b \tan ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\tan (c+d x)}{2 a d \left (a+b \tan ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 63, normalized size = 0.95 \begin {gather*} \frac {\frac {\text {ArcTan}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} \tan (c+d x)}{a+b \tan ^2(c+d x)}}{2 a^{3/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 55, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )}{2 a \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}+\frac {\arctan \left (\frac {b \tan \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\) | \(55\) |
default | \(\frac {\frac {\tan \left (d x +c \right )}{2 a \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}+\frac {\arctan \left (\frac {b \tan \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\) | \(55\) |
risch | \(\frac {i \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+a -b \right )}{a d \left (a -b \right ) \left (a \,{\mathrm e}^{4 i \left (d x +c \right )}-b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+a -b \right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d a}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d a}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 53, normalized size = 0.80 \begin {gather*} \frac {\frac {\tan \left (d x + c\right )}{a b \tan \left (d x + c\right )^{2} + a^{2}} + \frac {\arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs.
\(2 (54) = 108\).
time = 2.70, size = 327, normalized size = 4.95 \begin {gather*} \left [\frac {4 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + b\right )} \sqrt {-a b} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{3} - b \cos \left (d x + c\right )\right )} \sqrt {-a b} \sin \left (d x + c\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right )}{8 \, {\left (a^{2} b^{2} d + {\left (a^{3} b - a^{2} b^{2}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + b\right )} \sqrt {a b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - b\right )} \sqrt {a b}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{4 \, {\left (a^{2} b^{2} d + {\left (a^{3} b - a^{2} b^{2}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 70, normalized size = 1.06 \begin {gather*} \frac {\frac {\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {\tan \left (d x + c\right )}{{\left (b \tan \left (d x + c\right )^{2} + a\right )} a}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.14, size = 54, normalized size = 0.82 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{2\,a\,d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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