Optimal. Leaf size=144 \[ \frac {3 C \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x)}{d (3 m+4)}-\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right )}{d (2-3 m) (3 m+4) \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {20, 4046, 3772, 2643} \[ \frac {3 C \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x)}{d (3 m+4)}-\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right )}{d (2-3 m) (3 m+4) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {\sqrt [3]{b \sec (c+d x)} \int \sec ^{\frac {1}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (4+3 m)}+\frac {\left (\left (C \left (\frac {1}{3}+m\right )+A \left (\frac {4}{3}+m\right )\right ) \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {1}{3}+m}(c+d x) \, dx}{\left (\frac {4}{3}+m\right ) \sqrt [3]{\sec (c+d x)}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (4+3 m)}+\frac {\left (\left (C \left (\frac {1}{3}+m\right )+A \left (\frac {4}{3}+m\right )\right ) \cos ^{\frac {1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac {1}{3}-m}(c+d x) \, dx}{\frac {4}{3}+m}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (4+3 m)}-\frac {3 (C+3 C m+A (4+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (2-3 m) (4+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.85, size = 303, normalized size = 2.10 \[ -\frac {3 i 2^{m+\frac {4}{3}} e^{-\frac {1}{3} i (3 m+4) (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+\frac {4}{3}} \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {2 (A+2 C) e^{\frac {1}{3} i (3 m+7) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (-3 m-1);\frac {1}{6} (3 m+13);-e^{2 i (c+d x)}\right )}{3 m+7}+\frac {A e^{\frac {1}{3} i (3 m+1) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (-3 m-7);\frac {1}{6} (3 m+7);-e^{2 i (c+d x)}\right )}{3 m+1}+\frac {A e^{\frac {1}{3} i (3 m+13) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (5-3 m);\frac {1}{6} (3 m+19);-e^{2 i (c+d x)}\right )}{3 m+13}\right )}{d \sec ^{\frac {7}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{b \sec {\left (c + d x \right )}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________