Optimal. Leaf size=148 \[ -\frac {3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {3 \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 \sqrt {a^2+b^2} d}-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.28, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3593, 747, 827,
858, 221, 739, 212} \begin {gather*} -\frac {3 \left (2 a^2+b^2\right ) \sec (c+d x) \tanh ^{-1}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{2 b^4 d \sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}-\frac {3 a \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{b^4 d \sqrt {\sec ^2(c+d x)}}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 739
Rule 747
Rule 827
Rule 858
Rule 3593
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {\sec (c+d x) \text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{3/2}}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {(3 \sec (c+d x)) \text {Subst}\left (\int \frac {x \sqrt {1+\frac {x^2}{b^2}}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{2 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}-\frac {(3 \sec (c+d x)) \text {Subst}\left (\int \frac {-2+\frac {4 a x}{b^2}}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{4 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}-\frac {(3 a \sec (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b^5 d \sqrt {\sec ^2(c+d x)}}+\frac {\left (3 \left (1+\frac {2 a^2}{b^2}\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {3 a \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}-\frac {\left (3 \left (1+\frac {2 a^2}{b^2}\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{2 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {3 a \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{b^4 d \sqrt {\sec ^2(c+d x)}}-\frac {3 \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{2 b^4 \sqrt {a^2+b^2} d \sqrt {\sec ^2(c+d x)}}-\frac {\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(396\) vs. \(2(148)=296\).
time = 2.61, size = 396, normalized size = 2.68 \begin {gather*} \frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac {b^2 \left (a^2+b^2\right ) \sin (c+d x)}{a}+\frac {(2 a-b) b (2 a+b) (a \cos (c+d x)+b \sin (c+d x))}{a}+2 b (a \cos (c+d x)+b \sin (c+d x))^2+\frac {6 \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\sqrt {a^2+b^2}}+6 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2-6 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2+\frac {2 b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {2 b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{2 b^4 d (a+b \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 269, normalized size = 1.82
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \left (4 a^{4}-9 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}-\frac {b^{2} \left (13 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-2 a^{2} b +\frac {b^{3}}{2}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {3 \left (2 a^{2}+b^{2}\right ) \arctanh \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}-\frac {1}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {1}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}}{d}\) | \(269\) |
default | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (3 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \left (4 a^{4}-9 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}-\frac {b^{2} \left (13 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-2 a^{2} b +\frac {b^{3}}{2}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {3 \left (2 a^{2}+b^{2}\right ) \arctanh \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}-\frac {1}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {1}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}}{d}\) | \(269\) |
risch | \(\frac {-9 i a b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+9 i a b \,{\mathrm e}^{i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} d \,b^{3}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \,b^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{\sqrt {a^{2}+b^{2}}\, d \,b^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, d \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{\sqrt {a^{2}+b^{2}}\, d \,b^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, d \,b^{2}}\) | \(403\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs.
\(2 (138) = 276\).
time = 0.51, size = 518, normalized size = 3.50 \begin {gather*} \frac {\frac {2 \, {\left (6 \, a^{4} - a^{2} b^{2} + \frac {{\left (21 \, a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (6 \, a^{4} - 9 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, {\left (6 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (6 \, a^{4} - 9 \, a^{2} b^{2} + 2 \, b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (3 \, a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} b^{3} + \frac {4 \, a^{3} b^{4} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a^{3} b^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4 \, a^{3} b^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{4} b^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {{\left (3 \, a^{4} b^{3} - 4 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{4} b^{3} - 4 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {6 \, a \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{4}} + \frac {6 \, a \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{4}} - \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}}}{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. 513 vs.
\(2 (138) = 276\).
time = 0.47, size = 513, normalized size = 3.47 \begin {gather*} \frac {4 \, a^{2} b^{3} + 4 \, b^{5} + 6 \, {\left (2 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 18 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, {\left ({\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 6 \, {\left ({\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, {\left ({\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{4 \, {\left ({\left (a^{4} b^{4} - b^{8}\right )} d \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\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 ^{5}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (138) = 276\).
time = 0.69, size = 314, normalized size = 2.12 \begin {gather*} -\frac {\frac {6 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {6 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} + \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} + \frac {4}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} b^{3}} + \frac {2 \, {\left (3 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{4} + a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{2} a^{2} b^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.68, size = 1311, normalized size = 8.86 \begin {gather*} \frac {\frac {6\,a^2-b^2}{b^3}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a^4-9\,a^2\,b^2+b^4\right )}{a^2\,b^3}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (21\,a^2-2\,b^2\right )}{a\,b^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^4-9\,a^2\,b^2+2\,b^4\right )}{a^2\,b^3}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a^2-b^2\right )}{a\,b^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^2-2\,b^2\right )}{a\,b^2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^2-4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a^2-4\,b^2\right )-a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^2-8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {6\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{b^4\,d}+\frac {\mathrm {atan}\left (\frac {\frac {\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {288\,a^4}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,a^5\,b^3+108\,a^3\,b^5+9\,a\,b^7\right )}{b^9}-\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^3\,b^8+12\,a\,b^{10}\right )}{b^9}-48\,a^2+\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^3\,b^{11}+12\,a\,b^{13}\right )}{b^9}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )\,3{}\mathrm {i}}{2\,\left (a^2\,b^4+b^6\right )}+\frac {\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {288\,a^4}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,a^5\,b^3+108\,a^3\,b^5+9\,a\,b^7\right )}{b^9}-\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (48\,a^2-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^3\,b^8+12\,a\,b^{10}\right )}{b^9}+\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^3\,b^{11}+12\,a\,b^{13}\right )}{b^9}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )\,3{}\mathrm {i}}{2\,\left (a^2\,b^4+b^6\right )}}{\frac {16\,\left (54\,a^4+27\,a^2\,b^2\right )}{b^8}-\frac {16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (216\,a^5+108\,a^3\,b^2\right )}{b^9}-\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {288\,a^4}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,a^5\,b^3+108\,a^3\,b^5+9\,a\,b^7\right )}{b^9}-\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^3\,b^8+12\,a\,b^{10}\right )}{b^9}-48\,a^2+\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^3\,b^{11}+12\,a\,b^{13}\right )}{b^9}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}+\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (\frac {288\,a^4}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,a^5\,b^3+108\,a^3\,b^5+9\,a\,b^7\right )}{b^9}-\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (48\,a^2-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^3\,b^8+12\,a\,b^{10}\right )}{b^9}+\frac {3\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^3\,b^{11}+12\,a\,b^{13}\right )}{b^9}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}\right )}{2\,\left (a^2\,b^4+b^6\right )}}\right )\,\left (2\,a^2+b^2\right )\,\sqrt {a^2+b^2}\,3{}\mathrm {i}}{d\,\left (a^2\,b^4+b^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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