Optimal. Leaf size=92 \[ \frac {3 A b^2 \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3}}+\frac {3 (7 A+10 C) \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{40 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3012, 2643} \[ \frac {3 A b^2 \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3}}+\frac {3 (7 A+10 C) \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{40 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2643
Rule 3012
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx &=b^3 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{13/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3}}+\frac {1}{10} (b (7 A+10 C)) \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3}}+\frac {3 (7 A+10 C) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{40 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 91, normalized size = 0.99 \[ \frac {3 b^2 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \left (2 A \, _2F_1\left (-\frac {5}{3},\frac {1}{2};-\frac {2}{3};\cos ^2(c+d x)\right )+5 C \cos ^2(c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )\right )}{20 d (b \cos (c+d x))^{10/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3}}{b^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________