Optimal. Leaf size=513 \[ \frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{16 b^{9/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3} \]
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Rubi [A]
time = 0.39, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {3514, 3466,
3435, 3433, 3432, 3460, 3377, 2718, 3467, 3434} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) (d e-c f)^2 \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) (d e-c f)^2 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}+\frac {6 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {3 f (c+d x)^{4/3} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3460
Rule 3466
Rule 3467
Rule 3514
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {3 \text {Subst}\left (\int \left ((d e-c f)^2 x^2 \sin \left (a+b x^2\right )-2 f (-d e+c f) x^5 \sin \left (a+b x^2\right )+f^2 x^8 \sin \left (a+b x^2\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {\left (3 f^2\right ) \text {Subst}\left (\int x^8 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {(6 f (d e-c f)) \text {Subst}\left (\int x^5 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {\left (21 f^2\right ) \text {Subst}\left (\int x^6 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}+\frac {(3 f (d e-c f)) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {\left (105 f^2\right ) \text {Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d^3}+\frac {(6 f (d e-c f)) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b d^3}+\frac {\left (3 (d e-c f)^2 \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}-\frac {\left (3 (d e-c f)^2 \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {\left (315 f^2\right ) \text {Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d^3}-\frac {(6 f (d e-c f)) \text {Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b^2 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}+\frac {\left (315 f^2\right ) \text {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}+\frac {\left (315 f^2 \cos (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}+\frac {\left (315 f^2 \sin (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{16 b^{9/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.78, size = 510, normalized size = 0.99 \begin {gather*} -\frac {3 i \left ((\cos (a)+i \sin (a)) \left ((1+i) \left (-105 i f^2+8 b^3 (d e-c f)^2\right ) \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt [3]{c+d x}}{\sqrt {2}}\right )+2 \sqrt {b} \left (-105 f^2 \sqrt [3]{c+d x}-8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (8 d e-c f+7 d f x)+2 i b f (16 d e+19 c f+35 d f x)\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )\right )-\left (2 \sqrt {b} \left (-105 f^2 \sqrt [3]{c+d x}+8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (8 d e-c f+7 d f x)-2 i b f (16 d e+19 c f+35 d f x)\right )-(1+i) \left (105 i f^2+8 b^3 \left (d^2 e^2+c^2 f^2\right )\right ) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt [3]{c+d x}}{\sqrt {2}}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )+(8+8 i) b^3 c d e f \sqrt {2 \pi } \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt [3]{c+d x}}{\sqrt {2}}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )\right ) \left (\cos \left (a+b (c+d x)^{2/3}\right )-i \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{64 b^{9/2} d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 395, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {-\frac {3 f^{2} \left (d x +c \right )^{\frac {7}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {21 f^{2} \left (\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {5 \left (-\frac {\left (d x +c \right ) \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\frac {3 \left (d x +c \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{4 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{2 b}-\frac {3 \left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 \left (-2 c \,f^{2}+2 d e f \right ) \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}-\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{3}}\) | \(395\) |
default | \(\frac {-\frac {3 f^{2} \left (d x +c \right )^{\frac {7}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {21 f^{2} \left (\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {5 \left (-\frac {\left (d x +c \right ) \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\frac {3 \left (d x +c \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{4 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{2 b}-\frac {3 \left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 \left (-2 c \,f^{2}+2 d e f \right ) \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}-\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{3}}\) | \(395\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.36, size = 562, normalized size = 1.10 \begin {gather*} -\frac {3 \, {\left (\frac {8 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} c^{2} f^{2}}{b^{3} d^{2}} - \frac {16 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} c f e}{b^{3} d} + \frac {128 \, {\left (2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {4}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} c f^{2}}{b^{3} d^{2}} + \frac {8 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} e^{2}}{b^{3}} - \frac {128 \, {\left (2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {4}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} f e}{b^{3} d} - \frac {{\left (105 \, \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} - 16 \, {\left (4 \, {\left (d x + c\right )}^{\frac {7}{3}} b^{5} - 35 \, {\left (d x + c\right )} b^{3}\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) + 56 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{4} - 15 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2}\right )} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} f^{2}}{b^{6} d^{2}}\right )}}{128 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 314, normalized size = 0.61 \begin {gather*} \frac {3 \, {\left (\sqrt {2} {\left (105 \, \pi f^{2} \sin \left (a\right ) + 8 \, {\left (\pi b^{3} c^{2} f^{2} - 2 \, \pi b^{3} c d f e + \pi b^{3} d^{2} e^{2}\right )} \cos \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} {\left (105 \, \pi f^{2} \cos \left (a\right ) - 8 \, {\left (\pi b^{3} c^{2} f^{2} - 2 \, \pi b^{3} c d f e + \pi b^{3} d^{2} e^{2}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) + 4 \, {\left (35 \, b^{2} d f^{2} x + 19 \, b^{2} c f^{2} + 16 \, b^{2} d f e - 4 \, {\left (b^{4} d^{2} f^{2} x^{2} + 2 \, b^{4} d^{2} f x e + b^{4} d^{2} e^{2}\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - 2 \, {\left (105 \, {\left (d x + c\right )}^{\frac {1}{3}} b f^{2} - 4 \, {\left (7 \, b^{3} d f^{2} x - b^{3} c f^{2} + 8 \, b^{3} d f e\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{32 \, b^{5} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right )^{2} \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 4.48, size = 777, normalized size = 1.51 \begin {gather*} -\frac {3 \, {\left (f^{2} {\left (\frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (-8 i \, b^{3} c^{2} - 105\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b^{4} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {2 i \, {\left (8 i \, {\left (d x + c\right )}^{\frac {7}{3}} b^{3} - 16 i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{3} c + 8 i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{3} c^{2} - 28 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{2} + 32 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} c + 70 \, {\left (-i \, d x - i \, c\right )} b + 32 i \, b c + 105 \, {\left (d x + c\right )}^{\frac {1}{3}}\right )} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b^{4}}}{d^{2}} + \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (-8 i \, b^{3} c^{2} + 105\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b^{4} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {2 i \, {\left (8 i \, {\left (d x + c\right )}^{\frac {7}{3}} b^{3} - 16 i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{3} c + 8 i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{3} c^{2} + 28 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{2} - 32 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} c + 70 \, {\left (-i \, d x - i \, c\right )} b + 32 i \, b c - 105 \, {\left (d x + c\right )}^{\frac {1}{3}}\right )} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b^{4}}}{d^{2}}\right )} + 8 \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b}\right )} e^{2} - \frac {16 \, {\left (\frac {\sqrt {2} \sqrt {\pi } c \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {\sqrt {2} \sqrt {\pi } c \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {2 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c - 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b^{3}} + \frac {2 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b^{3}}\right )} f e}{d}\right )}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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