Optimal. Leaf size=95 \[ \frac {3 C (b \cos (c+d x))^{7/3} \sin (c+d x)}{10 b^2 d}-\frac {3 (10 A+7 C) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{70 b^2 d \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {16, 3093, 2722}
\begin {gather*} \frac {3 C \sin (c+d x) (b \cos (c+d x))^{7/3}}{10 b^2 d}-\frac {3 (10 A+7 C) \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right )}{70 b^2 d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int \cos (c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {\int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac {3 C (b \cos (c+d x))^{7/3} \sin (c+d x)}{10 b^2 d}+\frac {(10 A+7 C) \int (b \cos (c+d x))^{4/3} \, dx}{10 b}\\ &=\frac {3 C (b \cos (c+d x))^{7/3} \sin (c+d x)}{10 b^2 d}-\frac {3 (10 A+7 C) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{70 b^2 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 91, normalized size = 0.96 \begin {gather*} -\frac {3 (b \cos (c+d x))^{4/3} \cot (c+d x) \left (13 A \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right )+7 C \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{91 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.29, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________