Optimal. Leaf size=128 \[ \frac {\left (2 A b^2+a^2 (A+2 C)\right ) x}{2 a^3}-\frac {2 b \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b} d}-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d} \]
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Rubi [A]
time = 0.27, antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps
used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4190, 4189,
4004, 3916, 2738, 214} \begin {gather*} \frac {x \left (\frac {2 A b^2}{a^2}+A+2 C\right )}{2 a}-\frac {A b \sin (c+d x)}{a^2 d}-\frac {2 b \left (a^2 C+A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d \sqrt {a-b} \sqrt {a+b}}+\frac {A \sin (c+d x) \cos (c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4189
Rule 4190
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx &=\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (2 A b-a (A+2 C) \sec (c+d x)-A b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a}\\ &=-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {\int \frac {2 A b^2+a^2 (A+2 C)+a A b \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2}\\ &=\frac {\left (2 A b^2+a^2 (A+2 C)\right ) x}{2 a^3}-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\left (b \left (A b^2+a^2 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^3}\\ &=\frac {\left (2 A b^2+a^2 (A+2 C)\right ) x}{2 a^3}-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\left (A b^2+a^2 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^3}\\ &=\frac {\left (2 A b^2+a^2 (A+2 C)\right ) x}{2 a^3}-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\left (2 \left (A b^2+a^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 d}\\ &=\frac {\left (2 A b^2+a^2 (A+2 C)\right ) x}{2 a^3}-\frac {2 b \left (A b^2+a^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 \sqrt {a-b} \sqrt {a+b} d}-\frac {A b \sin (c+d x)}{a^2 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 115, normalized size = 0.90 \begin {gather*} \frac {2 \left (2 A b^2+a^2 (A+2 C)\right ) (c+d x)+\frac {8 b \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-4 a A b \sin (c+d x)+a^2 A \sin (2 (c+d x))}{4 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 160, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {-\frac {2 b \left (A \,b^{2}+a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(160\) |
default | \(\frac {-\frac {2 b \left (A \,b^{2}+a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(160\) |
risch | \(\frac {A x}{2 a}+\frac {x A \,b^{2}}{a^{3}}+\frac {x C}{a}+\frac {i A b \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}-\frac {i A b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, d a}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, d a}+\frac {A \sin \left (2 d x +2 c \right )}{4 a d}\) | \(380\) |
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 = 2.93, size = 386, normalized size = 3.02 \begin {gather*} \left [\frac {{\left ({\left (A + 2 \, C\right )} a^{4} + {\left (A - 2 \, C\right )} a^{2} b^{2} - 2 \, A b^{4}\right )} d x + {\left (C a^{2} b + A b^{3}\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 (2 \, A a^{3} b - 2 \, A a b^{3} - {\left (A a^{4} - A a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{5} - a^{3} b^{2}\right )} d}, \frac {{\left ({\left (A + 2 \, C\right )} a^{4} + {\left (A - 2 \, C\right )} a^{2} b^{2} - 2 \, A b^{4}\right )} d x - 2 \, {\left (C a^{2} b + A b^{3}\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 (2 \, A a^{3} b - 2 \, A a b^{3} - {\left (A a^{4} - A a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{5} - a^{3} b^{2}\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 ^{2}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 199, normalized size = 1.55 \begin {gather*} \frac {\frac {{\left (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )} {\left (d x + c\right )}}{a^{3}} - \frac {4 \, {\left (C a^{2} b + A b^{3}\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 )}}{\sqrt {-a^{2} + b^{2}} a^{3}} - \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.22, size = 2478, normalized size = 19.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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