Optimal. Leaf size=298 \[ -\frac{3}{16} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i b^2 \log ^2(f)}{12 f}+3 i d} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (b \log (f)+6 i f x)}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{-\frac{1}{12} i \left (\frac{b^2 \log ^2(f)}{f}+36 d\right )} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (-b \log (f)+6 i f x)}{\sqrt{6} \sqrt{f}}\right ) \]
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Rubi [A] time = 0.330467, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {4473, 2287, 2234, 2204, 2205} \[ -\frac{3}{16} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i b^2 \log ^2(f)}{12 f}+3 i d} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (b \log (f)+6 i f x)}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{-\frac{1}{12} i \left (\frac{b^2 \log ^2(f)}{f}+36 d\right )} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (-b \log (f)+6 i f x)}{\sqrt{6} \sqrt{f}}\right ) \]
Antiderivative was successfully verified.
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Rule 4473
Rule 2287
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int f^{a+b x} \cos ^3\left (d+f x^2\right ) \, dx &=\int \left (\frac{3}{8} e^{-i d-i f x^2} f^{a+b x}+\frac{3}{8} e^{i d+i f x^2} f^{a+b x}+\frac{1}{8} e^{-3 i d-3 i f x^2} f^{a+b x}+\frac{1}{8} e^{3 i d+3 i f x^2} f^{a+b x}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i f x^2} f^{a+b x} \, dx+\frac{1}{8} \int e^{3 i d+3 i f x^2} f^{a+b x} \, dx+\frac{3}{8} \int e^{-i d-i f x^2} f^{a+b x} \, dx+\frac{3}{8} \int e^{i d+i f x^2} f^{a+b x} \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i f x^2+a \log (f)+b x \log (f)} \, dx+\frac{1}{8} \int e^{3 i d+3 i f x^2+a \log (f)+b x \log (f)} \, dx+\frac{3}{8} \int e^{-i d-i f x^2+a \log (f)+b x \log (f)} \, dx+\frac{3}{8} \int e^{i d+i f x^2+a \log (f)+b x \log (f)} \, dx\\ &=\frac{1}{8} \left (e^{3 i d+\frac{i b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{-\frac{i (6 i f x+b \log (f))^2}{12 f}} \, dx+\frac{1}{8} \left (3 e^{-\frac{1}{4} i \left (4 d+\frac{b^2 \log ^2(f)}{f}\right )} f^a\right ) \int e^{\frac{i (-2 i f x+b \log (f))^2}{4 f}} \, dx+\frac{1}{8} \left (3 e^{\frac{1}{4} i \left (4 d+\frac{b^2 \log ^2(f)}{f}\right )} f^a\right ) \int e^{-\frac{i (2 i f x+b \log (f))^2}{4 f}} \, dx+\frac{1}{8} \left (e^{-\frac{1}{12} i \left (36 d+\frac{b^2 \log ^2(f)}{f}\right )} f^a\right ) \int e^{\frac{i (-6 i f x+b \log (f))^2}{12 f}} \, dx\\ &=-\frac{3}{16} \sqrt [4]{-1} e^{\frac{1}{4} i \left (4 d+\frac{b^2 \log ^2(f)}{f}\right )} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erf}\left (\frac{\sqrt [4]{-1} (2 i f x+b \log (f))}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) e^{3 i d+\frac{i b^2 \log ^2(f)}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{6}} \text{erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (6 i f x+b \log (f))}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} \sqrt [4]{-1} e^{-\frac{1}{4} i \left (4 d+\frac{b^2 \log ^2(f)}{f}\right )} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt [4]{-1} (2 i f x-b \log (f))}{2 \sqrt{f}}\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) e^{-\frac{1}{12} i \left (36 d+\frac{b^2 \log ^2(f)}{f}\right )} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{6}} \text{erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (6 i f x-b \log (f))}{\sqrt{6} \sqrt{f}}\right )\\ \end{align*}
Mathematica [A] time = 0.898799, size = 267, normalized size = 0.9 \[ \frac{1}{48} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{i b^2 \log ^2(f)}{4 f}} \left (9 e^{\frac{i b^2 \log ^2(f)}{2 f}} (\sin (d)-i \cos (d)) \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 f x-i b \log (f))}{2 \sqrt{f}}\right )+\sqrt{3} e^{\frac{i b^2 \log ^2(f)}{6 f}} \left (e^{\frac{i b^2 \log ^2(f)}{6 f}} (\sin (3 d)-i \cos (3 d)) \text{Erfi}\left (\frac{(1-i) b \log (f)+(6+6 i) f x}{2 \sqrt{6} \sqrt{f}}\right )-(\cos (3 d)-i \sin (3 d)) \text{Erfi}\left (\frac{(-1)^{3/4} (6 f x+i b \log (f))}{2 \sqrt{3} \sqrt{f}}\right )\right )-9 (\cos (d)-i \sin (d)) \text{Erfi}\left (\frac{(-1)^{3/4} (2 f x+i b \log (f))}{2 \sqrt{f}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.217, size = 235, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{3}{f}^{a}\sqrt{\pi }}{48}{{\rm e}^{{\frac{-{\frac{i}{12}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+36\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{3}\sqrt{if}x+{\frac{\ln \left ( f \right ) b\sqrt{3}}{6}{\frac{1}{\sqrt{if}}}} \right ){\frac{1}{\sqrt{if}}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{-{\frac{i}{4}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{if}}}} \right ){\frac{1}{\sqrt{if}}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{{\frac{i}{4}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{-if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-if}}}} \right ){\frac{1}{\sqrt{-if}}}}-{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{{\frac{i}{12}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+36\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{-3\,if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-3\,if}}}} \right ){\frac{1}{\sqrt{-3\,if}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.5599, size = 1517, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x} \cos ^{3}{\left (d + f x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.4692, size = 803, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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