Optimal. Leaf size=239 \[ -\frac {5 a \left (4 a^2+3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sqrt {a^2+b^2} \left (4 a^2+b^2\right ) \tanh ^{-1}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{2 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac {5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3593, 747, 827,
829, 858, 221, 739, 212} \begin {gather*} -\frac {5 \sqrt {a^2+b^2} \left (4 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^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 a \left (4 a^2+3 b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 b^6 d \sqrt {\sec ^2(c+d x)}}+\frac {5 \sec (c+d x) \left (4 a^2-2 a b \tan (c+d x)+b^2\right )}{2 b^5 d}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac {\sec ^5(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 829
Rule 858
Rule 3593
Rubi steps
\begin {align*} \int \frac {\sec ^7(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 )^{5/2}}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {x \left (1+\frac {x^2}{b^2}\right )^{3/2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{2 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {\left (-2+\frac {8 a x}{b^2}\right ) \sqrt {1+\frac {x^2}{b^2}}}{a+x} \, dx,x,b \tan (c+d x)\right )}{4 b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac {5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}-\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {-\frac {4 \left (2 a^2+b^2\right )}{b^4}+\frac {4 a \left (4 a^2+3 b^2\right ) x}{b^6}}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac {5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}+\frac {\left (5 \left (a^2+b^2\right ) \left (4 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^7 d \sqrt {\sec ^2(c+d x)}}-\frac {\left (5 a \left (4 a^2+3 b^2\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b^7 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {5 a \left (4 a^2+3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac {5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}-\frac {\left (5 \left (a^2+b^2\right ) \left (4 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^7 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {5 a \left (4 a^2+3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sqrt {a^2+b^2} \left (4 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^6 d \sqrt {\sec ^2(c+d x)}}-\frac {\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac {5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac {5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.61, size = 688, normalized size = 2.88 \begin {gather*} \frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac {6 b^2 \left (a^2+b^2\right )^2 \sin (c+d x)}{a}+\frac {6 (a-i b) (a+i b) b \left (8 a^2-b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}{a}+2 b \left (36 a^2+13 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt {a^2+b^2} \left (4 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+30 a \left (4 a^2+3 b^2\right ) \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))^2-30 a \left (4 a^2+3 b^2\right ) \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 {b^2 (-9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b \left (36 a^2+13 b^2\right ) \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^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2 b \left (36 a^2+13 b^2\right ) \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 )}{12 b^6 d (a+b \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(443\) vs.
\(2(219)=438\).
time = 0.53, size = 444, normalized size = 1.86 Too large to display
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. 902 vs.
\(2 (223) = 446\).
time = 0.52, size = 902, normalized size = 3.77 \begin {gather*} \frac {\frac {2 \, {\left (60 \, a^{6} + 35 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + \frac {{\left (210 \, a^{5} b + 125 \, a^{3} b^{3} - 6 \, a b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (120 \, a^{6} - 10 \, a^{4} b^{2} - 55 \, a^{2} b^{4} + 3 \, b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (330 \, a^{5} b + 205 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, {\left (180 \, a^{6} - 95 \, a^{4} b^{2} - 120 \, a^{2} b^{4} + 9 \, b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, {\left (60 \, a^{5} b + 35 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6 \, {\left (40 \, a^{6} - 30 \, a^{4} b^{2} - 35 \, a^{2} b^{4} + 3 \, b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, {\left (50 \, a^{5} b + 25 \, a^{3} b^{3} - 4 \, a b^{5}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3 \, {\left (20 \, a^{6} - 15 \, a^{4} b^{2} - 15 \, a^{2} b^{4} + 2 \, b^{6}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3 \, {\left (10 \, a^{5} b + 5 \, a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}}{a^{4} b^{5} + \frac {4 \, a^{3} b^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, a^{3} b^{6} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {24 \, a^{3} b^{6} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {16 \, a^{3} b^{6} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {4 \, a^{3} b^{6} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{4} b^{5} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {{\left (5 \, a^{4} b^{5} - 4 \, a^{2} b^{7}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, {\left (5 \, a^{4} b^{5} - 6 \, a^{2} b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, {\left (5 \, a^{4} b^{5} - 6 \, a^{2} b^{7}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (5 \, a^{4} b^{5} - 4 \, a^{2} b^{7}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{6}} + \frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{6}} - \frac {15 \, {\left (4 \, a^{4} + 5 \, a^{2} b^{2} + b^{4}\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^{6}}}{6 \, 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. 564 vs.
\(2 (223) = 446\).
time = 0.51, size = 564, normalized size = 2.36 \begin {gather*} \frac {4 \, b^{5} + 30 \, {\left (4 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (4 \, a^{4} - 3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3}\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 ) - 15 \, {\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (a b^{4} \cos \left (d x + c\right ) - 6 \, {\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (2 \, a b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + b^{8} d \cos \left (d x + c\right )^{3} + {\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{5}\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 ^{7}{\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. 510 vs.
\(2 (223) = 446\).
time = 0.74, size = 510, normalized size = 2.13 \begin {gather*} -\frac {\frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {15 \, {\left (4 \, a^{4} + 5 \, a^{2} b^{2} + b^{4}\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^{6}} + \frac {2 \, {\left (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 72 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} + 14 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} b^{5}} + \frac {6 \, {\left (7 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 25 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 23 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{6} - 7 \, a^{4} b^{2} + a^{2} b^{4}\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^{5}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.94, size = 1203, normalized size = 5.03 \begin {gather*} \frac {\frac {60\,a^4+35\,a^2\,b^2-3\,b^4}{3\,b^5}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (210\,a^4+125\,a^2\,b^2-6\,b^4\right )}{3\,a\,b^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (20\,a^6-15\,a^4\,b^2-15\,a^2\,b^4+2\,b^6\right )}{a^2\,b^5}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (40\,a^6-30\,a^4\,b^2-35\,a^2\,b^4+3\,b^6\right )}{a^2\,b^5}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (120\,a^6-10\,a^4\,b^2-55\,a^2\,b^4+3\,b^6\right )}{3\,a^2\,b^5}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (180\,a^6-95\,a^4\,b^2-120\,a^2\,b^4+9\,b^6\right )}{3\,a^2\,b^5}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (10\,a^4+5\,a^2\,b^2-2\,b^4\right )}{a\,b^4}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (50\,a^4+25\,a^2\,b^2-4\,b^4\right )}{a\,b^4}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (60\,a^4+35\,a^2\,b^2-3\,b^4\right )}{a\,b^4}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (330\,a^4+205\,a^2\,b^2-12\,b^4\right )}{3\,a\,b^4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (5\,a^2-4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (5\,a^2-4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (10\,a^2-12\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (10\,a^2-12\,b^2\right )-a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+a^2-16\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-16\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {atanh}\left (\frac {3000\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3000\,a^2+\frac {7000\,a^4}{b^2}+\frac {4000\,a^6}{b^4}}+\frac {7000\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7000\,a^4+3000\,a^2\,b^2+\frac {4000\,a^6}{b^2}}+\frac {4000\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4000\,a^6+7000\,a^4\,b^2+3000\,a^2\,b^4}\right )\,\left (20\,a^3+15\,a\,b^2\right )}{b^6\,d}+\frac {5\,\mathrm {atanh}\left (\frac {1000\,a^2\,\sqrt {a^2+b^2}}{1000\,a^2\,b+\frac {5000\,a^4}{b}+\frac {4000\,a^6}{b^3}+10000\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+2000\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {8000\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b^2}}+\frac {4000\,a^4\,\sqrt {a^2+b^2}}{5000\,a^4\,b+1000\,a^2\,b^3+\frac {4000\,a^6}{b}+8000\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+2000\,a\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+10000\,a^3\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {9000\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}}{5000\,a^4+1000\,a^2\,b^2+\frac {4000\,a^6}{b^2}+2000\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+10000\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {8000\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b}}+\frac {4000\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}}{4000\,a^6+8000\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5\,b+5000\,a^4\,b^2+10000\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^3+1000\,a^2\,b^4+2000\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^5}+\frac {2000\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}}{1000\,a^2+\frac {5000\,a^4}{b^2}+\frac {4000\,a^6}{b^4}+\frac {10000\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b}+\frac {8000\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{b^3}+2000\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\left (4\,a^2+b^2\right )\,\sqrt {a^2+b^2}}{b^6\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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