Optimal. Leaf size=180 \[ -\frac {(2 b B-a C) x}{a^3}+\frac {2 b \left (3 a^2 b B-2 b^3 B-2 a^3 C+a b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.46, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4115,
4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {x (2 b B-a C)}{a^3}+\frac {\left (a^2 B+a b C-2 b^2 B\right ) \sin (c+d x)}{a^2 d \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {2 b \left (-2 a^3 C+3 a^2 b B+a b^2 C-2 b^3 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{3/2} (a+b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4115
Rule 4157
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos (c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\\ &=\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-a^2 B+2 b^2 B-a b C+a (b B-a C) \sec (c+d x)-b (b B-a C) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {-\left (a^2-b^2\right ) (2 b B-a C)+a b (b B-a C) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {(2 b B-a C) x}{a^3}+\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (b \left (3 a^2 b B-2 b^3 B-2 a^3 C+a b^2 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )}\\ &=-\frac {(2 b B-a C) x}{a^3}+\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 a^2 b B-2 b^3 B-2 a^3 C+a b^2 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^3 \left (a^2-b^2\right )}\\ &=-\frac {(2 b B-a C) x}{a^3}+\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (2 \left (3 a^2 b B-2 b^3 B-2 a^3 C+a b^2 C\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 )}{a^3 \left (a^2-b^2\right ) d}\\ &=-\frac {(2 b B-a C) x}{a^3}+\frac {2 b \left (3 a^2 b B-2 b^3 B-2 a^3 C+a 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^3 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2 B-2 b^2 B+a b C\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.88, size = 147, normalized size = 0.82 \begin {gather*} \frac {(-2 b B+a C) (c+d x)+\frac {2 b \left (-3 a^2 b B+2 b^3 B+2 a^3 C-a b^2 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+a B \sin (c+d x)+\frac {a b^2 (-b B+a C) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 213, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {B a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (2 b B -a C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}-\frac {2 b \left (-\frac {a b \left (b B -a C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {\left (3 a^{2} b B -2 b^{3} B -2 a^{3} C +C \,b^{2} a \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}}{d}\) | \(213\) |
default | \(\frac {-\frac {2 \left (-\frac {B a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (2 b B -a C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}-\frac {2 b \left (-\frac {a b \left (b B -a C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {\left (3 a^{2} b B -2 b^{3} B -2 a^{3} C +C \,b^{2} a \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}}{d}\) | \(213\) |
risch | \(-\frac {2 x b B}{a^{3}}+\frac {x C}{a^{2}}-\frac {i B \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}+\frac {i B \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}+\frac {2 i b^{2} \left (-b B +a C \right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{3} \left (a^{2}-b^{2}\right ) d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\) | \(810\) |
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. 364 vs.
\(2 (171) = 342\).
time = 3.07, size = 788, normalized size = 4.38 \begin {gather*} \left [\frac {2 \, {\left (C a^{6} - 2 \, B a^{5} b - 2 \, C a^{4} b^{2} + 4 \, B a^{3} b^{3} + C a^{2} b^{4} - 2 \, B a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (C a^{5} b - 2 \, B a^{4} b^{2} - 2 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + C a b^{5} - 2 \, B b^{6}\right )} d x + {\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - C a b^{4} + 2 \, B b^{5} + {\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} - C a^{2} b^{3} + 2 \, B a 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 ) - {\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 ) + 2 \, {\left (B a^{5} b + C a^{4} b^{2} - 3 \, B a^{3} b^{3} - C a^{2} b^{4} + 2 \, B a b^{5} + {\left (B a^{6} - 2 \, B a^{4} b^{2} + B a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d\right )}}, \frac {{\left (C a^{6} - 2 \, B a^{5} b - 2 \, C a^{4} b^{2} + 4 \, B a^{3} b^{3} + C a^{2} b^{4} - 2 \, B a b^{5}\right )} d x \cos \left (d x + c\right ) + {\left (C a^{5} b - 2 \, B a^{4} b^{2} - 2 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + C a b^{5} - 2 \, B b^{6}\right )} d x - {\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - C a b^{4} + 2 \, B b^{5} + {\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} - C a^{2} b^{3} + 2 \, B a b^{4}\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 (B a^{5} b + C a^{4} b^{2} - 3 \, B a^{3} b^{3} - C a^{2} b^{4} + 2 \, B a b^{5} + {\left (B a^{6} - 2 \, B a^{4} b^{2} + B a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d}\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 (B + C \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )} \sec {\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. 1107 vs.
\(2 (171) = 342\).
time = 0.59, size = 1107, normalized size = 6.15 \begin {gather*} \frac {\frac {{\left (C a^{8} - 2 \, B a^{7} b - 3 \, C a^{7} b + 5 \, B a^{6} b^{2} - 2 \, C a^{6} b^{2} + 4 \, B a^{5} b^{3} + 5 \, C a^{5} b^{3} - 9 \, B a^{4} b^{4} + C a^{4} b^{4} - 2 \, B a^{3} b^{5} - 2 \, C a^{3} b^{5} + 4 \, B a^{2} b^{6} - C a^{3} {\left | -a^{5} + a^{3} b^{2} \right |} + 2 \, B a^{2} b {\left | -a^{5} + a^{3} b^{2} \right |} - C a^{2} b {\left | -a^{5} + a^{3} b^{2} \right |} + B a b^{2} {\left | -a^{5} + a^{3} b^{2} \right |} + C a b^{2} {\left | -a^{5} + a^{3} b^{2} \right |} - 2 \, B b^{3} {\left | -a^{5} + a^{3} b^{2} \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{4} b - a^{2} b^{3} + \sqrt {{\left (a^{5} + a^{4} b - a^{3} b^{2} - a^{2} b^{3}\right )} {\left (a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}\right )} + {\left (a^{4} b - a^{2} b^{3}\right )}^{2}}}{a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}}}}\right )\right )}}{a^{4} b {\left | -a^{5} + a^{3} b^{2} \right |} - a^{2} b^{3} {\left | -a^{5} + a^{3} b^{2} \right |} + {\left (a^{5} - a^{3} b^{2}\right )}^{2}} - \frac {{\left ({\left (2 \, a^{2} b + a b^{2} - 2 \, b^{3}\right )} \sqrt {-a^{2} + b^{2}} B {\left | -a^{5} + a^{3} b^{2} \right |} {\left | -a + b \right |} - {\left (a^{3} + a^{2} b - a b^{2}\right )} \sqrt {-a^{2} + b^{2}} C {\left | -a^{5} + a^{3} b^{2} \right |} {\left | -a + b \right |} + {\left (2 \, a^{7} b - 5 \, a^{6} b^{2} - 4 \, a^{5} b^{3} + 9 \, a^{4} b^{4} + 2 \, a^{3} b^{5} - 4 \, a^{2} b^{6}\right )} \sqrt {-a^{2} + b^{2}} B {\left | -a + b \right |} - {\left (a^{8} - 3 \, a^{7} b - 2 \, a^{6} b^{2} + 5 \, a^{5} b^{3} + a^{4} b^{4} - 2 \, a^{3} b^{5}\right )} \sqrt {-a^{2} + b^{2}} C {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{4} b - a^{2} b^{3} - \sqrt {{\left (a^{5} + a^{4} b - a^{3} b^{2} - a^{2} b^{3}\right )} {\left (a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}\right )} + {\left (a^{4} b - a^{2} b^{3}\right )}^{2}}}{a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}}}}\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )}^{2} {\left (a^{2} - 2 \, a b + b^{2}\right )} - {\left (a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{3} b^{4} - a^{2} b^{5}\right )} {\left | -a^{5} + a^{3} b^{2} \right |}} + \frac {2 \, {\left (B 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} - B 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} + 2 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B 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 ) + B 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 ) + 2 \, B 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 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} {\left (a^{4} - a^{2} b^{2}\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.05, size = 2500, normalized size = 13.89 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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