Optimal. Leaf size=127 \[ \frac {(5 a+b) (a-b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2} d}-\frac {(a-b)^3 \tan (c+d x)}{2 a b^3 d \left (a+b \tan ^2(c+d x)\right )}-\frac {(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3675, 390, 385, 205} \[ \frac {(5 a+b) (a-b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2} d}-\frac {(a-b)^3 \tan (c+d x)}{2 a b^3 d \left (a+b \tan ^2(c+d x)\right )}-\frac {(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 390
Rule 3675
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {2 a-3 b}{b^3}+\frac {x^2}{b^2}+\frac {(a-b)^2 (2 a+b)+3 (a-b)^2 b x^2}{b^3 \left (a+b x^2\right )^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}+\frac {\operatorname {Subst}\left (\int \frac {(a-b)^2 (2 a+b)+3 (a-b)^2 b x^2}{\left (a+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b^3 d}\\ &=-\frac {(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}-\frac {(a-b)^3 \tan (c+d x)}{2 a b^3 d \left (a+b \tan ^2(c+d x)\right )}+\frac {\left ((a-b)^2 (5 a+b)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{2 a b^3 d}\\ &=\frac {(a-b)^2 (5 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2} d}-\frac {(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac {\tan ^3(c+d x)}{3 b^2 d}-\frac {(a-b)^3 \tan (c+d x)}{2 a b^3 d \left (a+b \tan ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 135, normalized size = 1.06 \[ \frac {\frac {3 (a-b)^2 (5 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+4 \sqrt {b} (4 b-3 a) \tan (c+d x)+\frac {3 \sqrt {b} (b-a)^3 \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 b^{3/2} \tan (c+d x) \sec ^2(c+d x)}{6 b^{7/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 597, normalized size = 4.70 \[ \left [-\frac {3 \, {\left ({\left (5 \, a^{4} - 14 \, a^{3} b + 12 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b - 9 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3}\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 ) - 4 \, {\left (2 \, a^{2} b^{3} - {\left (15 \, a^{4} b - 37 \, a^{3} b^{2} + 25 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{2} b^{5} d \cos \left (d x + c\right )^{3} + {\left (a^{3} b^{4} - a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{5}\right )}}, -\frac {3 \, {\left ({\left (5 \, a^{4} - 14 \, a^{3} b + 12 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b - 9 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3}\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 ) - 2 \, {\left (2 \, a^{2} b^{3} - {\left (15 \, a^{4} b - 37 \, a^{3} b^{2} + 25 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} b^{5} d \cos \left (d x + c\right )^{3} + {\left (a^{3} b^{4} - a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.29, size = 180, normalized size = 1.42 \[ \frac {\frac {3 \, {\left (5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )\right )}}{\sqrt {a b} a b^{3}} - \frac {3 \, {\left (a^{3} \tan \left (d x + c\right ) - 3 \, a^{2} b \tan \left (d x + c\right ) + 3 \, a b^{2} \tan \left (d x + c\right ) - b^{3} \tan \left (d x + c\right )\right )}}{{\left (b \tan \left (d x + c\right )^{2} + a\right )} a b^{3}} + \frac {2 \, {\left (b^{4} \tan \left (d x + c\right )^{3} - 6 \, a b^{3} \tan \left (d x + c\right ) + 9 \, b^{4} \tan \left (d x + c\right )\right )}}{b^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.92, size = 275, normalized size = 2.17 \[ \frac {\tan ^{3}\left (d x +c \right )}{3 b^{2} d}-\frac {2 a \tan \left (d x +c \right )}{d \,b^{3}}+\frac {3 \tan \left (d x +c \right )}{b^{2} d}-\frac {a^{2} \tan \left (d x +c \right )}{2 d \,b^{3} \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}+\frac {3 a \tan \left (d x +c \right )}{2 d \,b^{2} \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}-\frac {3 \tan \left (d x +c \right )}{2 d b \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}+\frac {\tan \left (d x +c \right )}{2 a d \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}+\frac {5 a^{2} \arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right )}{2 d \,b^{3} \sqrt {a b}}-\frac {9 a \arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right )}{2 d \,b^{2} \sqrt {a b}}+\frac {3 \arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right )}{2 d b \sqrt {a b}}+\frac {\arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right )}{2 d a \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 137, normalized size = 1.08 \[ -\frac {\frac {3 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \tan \left (d x + c\right )}{a b^{4} \tan \left (d x + c\right )^{2} + a^{2} b^{3}} - \frac {2 \, {\left (b \tan \left (d x + c\right )^{3} - 3 \, {\left (2 \, a - 3 \, b\right )} \tan \left (d x + c\right )\right )}}{b^{3}} - \frac {3 \, {\left (5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.17, size = 167, normalized size = 1.31 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,b^2\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {2\,a}{b^3}-\frac {3}{b^2}\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,a\,d\,\left (b^4\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (c+d\,x\right )\,{\left (a-b\right )}^2\,\left (5\,a+b\right )}{\sqrt {a}\,\left (5\,a^3-9\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,{\left (a-b\right )}^2\,\left (5\,a+b\right )}{2\,a^{3/2}\,b^{7/2}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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