Optimal. Leaf size=177 \[ \frac {(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {2 \left (A b^4+a^3 b B-2 a b^3 B-2 a^4 C+3 a^2 b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.45, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4175, 4167,
4083, 3855, 3916, 2738, 214} \begin {gather*} \frac {a \tan (c+d x) \left (A b^2-a (b B-a C)\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {2 \left (-2 a^4 C+a^3 b B+3 a^2 b^2 C-2 a b^3 B+A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {C \tan (c+d x)}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4167
Rule 4175
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-b \left (A b^2-a (b B-a C)\right )+\left (a^2-b^2\right ) (b B-a C) \sec (c+d x)+b \left (a^2-b^2\right ) C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-b^2 \left (A b^2-a (b B-a C)\right )+b \left (a^2-b^2\right ) (b B-2 a C) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {(b B-2 a C) \int \sec (c+d x) \, dx}{b^3}-\frac {\left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=\frac {(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^4 \left (a^2-b^2\right )}\\ &=\frac {(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (2 \left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 \left (a^2-b^2\right ) d}\\ &=\frac {(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {2 \left (A b^4+a^3 b B-2 a b^3 B-2 a^4 C+3 a^2 b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(382\) vs. \(2(177)=354\).
time = 3.20, size = 382, normalized size = 2.16 \begin {gather*} \frac {2 (b+a \cos (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))}{\left (a^2-b^2\right )^{3/2}}-(b B-2 a C) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+(b B-2 a C) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {a b \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{(a-b) (a+b)}\right )}{b^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.85, size = 261, normalized size = 1.47 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 550 vs.
\(2 (171) = 342\).
time = 54.36, size = 1156, normalized size = 6.53 \begin {gather*} \left [\frac {{\left ({\left (2 \, C a^{5} - B a^{4} b - 3 \, C a^{3} b^{2} + 2 \, B a^{2} b^{3} - A a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{4} b - B a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\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 ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left ({\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6} + {\left (2 \, C a^{5} b - B a^{4} b^{2} + {\left (A - 3 \, C\right )} a^{3} b^{3} + B a^{2} b^{4} - {\left (A - C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )\right )}}, \frac {2 \, {\left ({\left (2 \, C a^{5} - B a^{4} b - 3 \, C a^{3} b^{2} + 2 \, B a^{2} b^{3} - A a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{4} b - B a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left ({\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6} + {\left (2 \, C a^{5} b - B a^{4} b^{2} + {\left (A - 3 \, C\right )} a^{3} b^{3} + B a^{2} b^{4} - {\left (A - C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )\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 {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 443 vs.
\(2 (171) = 342\).
time = 0.50, size = 443, normalized size = 2.50 \begin {gather*} \frac {\frac {2 \, {\left (2 \, C a^{4} - B a^{3} b - 3 \, C a^{2} b^{2} + 2 \, B a b^{3} - A b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {2 \, {\left (2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} {\left (a^{2} b^{2} - b^{4}\right )}} - \frac {{\left (2 \, C a - B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} + \frac {{\left (2 \, C a - B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.16, size = 2500, normalized size = 14.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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