Optimal. Leaf size=171 \[ -\frac {2 A b x}{a^3}+\frac {2 \left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.31, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4186, 4189,
4004, 3916, 2738, 214} \begin {gather*} -\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 d \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {2 \left (a^4 C+3 a^2 A b^2-2 A b^4\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 4186
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\frac {\left (A b^2+a^2 C\right ) \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 (2 A b^2-a^2 (A-C)+a b (A+C) \sec (c+d x)-\left (A b^2+a^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {-2 A b \left (a^2-b^2\right )+a \left (A b^2+a^2 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^3 b \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \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 A b^2-2 A b^4+a^4 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 b \left (a^2-b^2\right ) d}\\ &=-\frac {2 A b x}{a^3}+\frac {2 \left (3 a^2 A b^2-2 A b^4+a^4 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 (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \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 = 1.04, size = 137, normalized size = 0.80 \begin {gather*} \frac {-2 A b (c+d x)-\frac {2 \left (3 a^2 A b^2-2 A b^4+a^4 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 A \sin (c+d x)-\frac {a b \left (A b^2+a^2 C\right ) \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 = 0.22, size = 203, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {a b \left (A \,b^{2}+a^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {\left (3 a^{2} A \,b^{2}-2 A \,b^{4}+a^{4} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}}{d}\) | \(203\) |
default | \(\frac {-\frac {2 \left (-\frac {a b \left (A \,b^{2}+a^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {\left (3 a^{2} A \,b^{2}-2 A \,b^{4}+a^{4} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}}{d}\) | \(203\) |
risch | \(-\frac {2 A b x}{a^{3}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}-\frac {2 i b \left (A \,b^{2}+a^{2} C \right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{3} \left (a^{2}-b^{2}\right ) d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A \,b^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A \,b^{4}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A \,b^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A \,b^{4}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}\) | \(631\) |
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 [A]
time = 3.84, size = 642, normalized size = 3.75 \begin {gather*} \left [-\frac {4 \, {\left (A a^{5} b - 2 \, A a^{3} b^{3} + A a b^{5}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (A a^{4} b^{2} - 2 \, A a^{2} b^{4} + A b^{6}\right )} d x + {\left (C a^{4} b + 3 \, A a^{2} b^{3} - 2 \, A b^{5} + {\left (C a^{5} + 3 \, A a^{3} b^{2} - 2 \, A 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 ({\left (A - C\right )} a^{5} b - {\left (3 \, A - C\right )} a^{3} b^{3} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A 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 {2 \, {\left (A a^{5} b - 2 \, A a^{3} b^{3} + A a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (A a^{4} b^{2} - 2 \, A a^{2} b^{4} + A b^{6}\right )} d x - {\left (C a^{4} b + 3 \, A a^{2} b^{3} - 2 \, A b^{5} + {\left (C a^{5} + 3 \, A a^{3} b^{2} - 2 \, A 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 ({\left (A - C\right )} a^{5} b - {\left (3 \, A - C\right )} a^{3} b^{3} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A 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 (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\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. 988 vs.
\(2 (160) = 320\).
time = 0.53, size = 988, normalized size = 5.78 \begin {gather*} \frac {\frac {{\left (C a^{8} - 2 \, A a^{7} b + 5 \, A a^{6} b^{2} - C a^{6} b^{2} + 4 \, A a^{5} b^{3} - 9 \, A a^{4} b^{4} - 2 \, A a^{3} b^{5} + 4 \, A a^{2} b^{6} + C a^{3} {\left | -a^{5} + a^{3} b^{2} \right |} + 2 \, A a^{2} b {\left | -a^{5} + a^{3} b^{2} \right |} + A a b^{2} {\left | -a^{5} + a^{3} b^{2} \right |} - 2 \, A 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 (\sqrt {-a^{2} + b^{2}} C a^{3} {\left | -a^{5} + a^{3} b^{2} \right |} {\left | -a + b \right |} + {\left (2 \, a^{2} b + a b^{2} - 2 \, b^{3}\right )} \sqrt {-a^{2} + b^{2}} A {\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}} A {\left | -a + b \right |} - {\left (a^{8} - a^{6} b^{2}\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 (A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A 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} + 2 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A 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 ) + 2 \, A 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 = 11.67, size = 2500, normalized size = 14.62 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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