Optimal. Leaf size=219 \[ \frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.6097, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {3044, 2975, 2980, 2772, 2771} \[ \frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3044
Rule 2975
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{3 a A}{2}+\frac{1}{2} a (4 A+9 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac{2 a A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{4} a^2 (52 A+63 C)+\frac{1}{4} a^2 (40 A+63 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{105} (a (136 A+189 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{315} (2 a (136 A+189 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.718353, size = 123, normalized size = 0.56 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((748 A+567 C) \cos (c+d x)+(748 A+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+136 A \cos (4 (c+d x))+752 A+189 C \cos (3 (c+d x))+189 C \cos (4 (c+d x))+693 C)}{630 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.117, size = 122, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 272\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+378\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+136\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+189\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+102\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+63\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+85\,A\cos \left ( dx+c \right ) +35\,A \right ) }{315\,d\sin \left ( dx+c \right ) }\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.78124, size = 711, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46071, size = 324, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (2 \,{\left (136 \, A + 189 \, C\right )} a \cos \left (d x + c\right )^{4} +{\left (136 \, A + 189 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (34 \, A + 21 \, C\right )} a \cos \left (d x + c\right )^{2} + 85 \, A a \cos \left (d x + c\right ) + 35 \, A a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]