Optimal. Leaf size=192 \[ \frac {5 a^{5/2} (5 A+8 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a \sec (c+d x)+a}}-\frac {a^2 (3 A-8 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d}+\frac {5 a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d} \]
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Rubi [A] time = 0.62, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4087, 4017, 4018, 4015, 3774, 203} \[ \frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a \sec (c+d x)+a}}-\frac {a^2 (3 A-8 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {5 a^{5/2} (5 A+8 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d}+\frac {5 a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 4015
Rule 4017
Rule 4018
Rule 4087
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {5 a A}{2}-\frac {1}{2} a (A-6 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (31 A+24 C)-\frac {3}{4} a^2 (3 A-8 C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (49 A-24 C)+\frac {1}{8} a^3 (13 A+72 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{16} \left (5 a^2 (5 A+8 C)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}-\frac {\left (5 a^3 (5 A+8 C)\right ) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}\\ &=\frac {5 a^{5/2} (5 A+8 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 132, normalized size = 0.69 \[ \frac {a^2 \sin (c+d x) \sqrt {a (\sec (c+d x)+1)} \left (\sqrt {\sec (c+d x)-1} (3 (27 A+8 C) \cos (c+d x)+17 A \cos (2 (c+d x))+2 A \cos (3 (c+d x))+17 A+48 C)+15 (5 A+8 C) \tan ^{-1}\left (\sqrt {\sec (c+d x)-1}\right )\right )}{24 d (\cos (c+d x)+1) \sqrt {\sec (c+d x)-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 380, normalized size = 1.98 \[ \left [\frac {15 \, {\left ({\left (5 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (5 \, A + 8 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 34 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (25 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (5 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (5 \, A + 8 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 34 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (25 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.89, size = 967, normalized size = 5.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.50, size = 583, normalized size = 3.04 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (75 A \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+120 C \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+150 A \sqrt {2}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+240 C \sqrt {2}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+75 A \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )+120 C \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )+64 A \left (\cos ^{6}\left (d x +c \right )\right )+208 A \left (\cos ^{5}\left (d x +c \right )\right )+328 A \left (\cos ^{4}\left (d x +c \right )\right )+192 C \left (\cos ^{4}\left (d x +c \right )\right )-600 A \left (\cos ^{3}\left (d x +c \right )\right )+192 C \left (\cos ^{3}\left (d x +c \right )\right )-384 C \left (\cos ^{2}\left (d x +c \right )\right )\right ) a^{2}}{192 d \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^3\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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