Optimal. Leaf size=142 \[ \frac{1}{4} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\frac{1}{4} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+2 i f x)}{2 \sqrt{f}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.210487, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4472, 2287, 2234, 2204, 2205} \[ \frac{1}{4} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+2 i f x)}{2 \sqrt{f}}\right )-\frac{1}{4} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f}+4 d\right )} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+2 i f x)}{2 \sqrt{f}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4472
Rule 2287
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int f^{a+b x} \sin \left (d+f x^2\right ) \, dx &=\int \left (\frac{1}{2} i e^{-i d-i f x^2} f^{a+b x}-\frac{1}{2} i e^{i d+i f x^2} f^{a+b x}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i d-i f x^2} f^{a+b x} \, dx-\frac{1}{2} i \int e^{i d+i f x^2} f^{a+b x} \, dx\\ &=\frac{1}{2} i \int e^{-i d-i f x^2+a \log (f)+b x \log (f)} \, dx-\frac{1}{2} i \int e^{i d+i f x^2+a \log (f)+b x \log (f)} \, dx\\ &=\frac{1}{2} \left (i 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}{2} \left (i 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}{4} (-1)^{3/4} 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 )-\frac{1}{4} (-1)^{3/4} 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 )\\ \end{align*}
Mathematica [A] time = 0.2288, size = 132, normalized size = 0.93 \[ -\frac{1}{4} \sqrt [4]{-1} \sqrt{\pi } f^{a-\frac{1}{2}} e^{-\frac{i b^2 \log ^2(f)}{4 f}} \left (e^{\frac{i b^2 \log ^2(f)}{2 f}} (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 f x-i b \log (f))}{2 \sqrt{f}}\right )+(\sin (d)+i \cos (d)) \text{Erfi}\left (\frac{(-1)^{3/4} (2 f x+i b \log (f))}{2 \sqrt{f}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.267, size = 116, normalized size = 0.8 \begin{align*}{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\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{i}{4}}{f}^{a}\sqrt{\pi }{{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.504809, size = 749, normalized size = 5.27 \begin{align*} \frac{i \, \sqrt{2} \pi \sqrt{\frac{f}{\pi }} e^{\left (\frac{-i \, b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 i \, d f}{4 \, f}\right )} \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, f x + i \, b \log \left (f\right )\right )} \sqrt{\frac{f}{\pi }}}{2 \, f}\right ) + i \, \sqrt{2} \pi \sqrt{\frac{f}{\pi }} e^{\left (\frac{i \, b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) + 4 i \, d f}{4 \, f}\right )} \operatorname{C}\left (-\frac{\sqrt{2}{\left (2 \, f x - i \, b \log \left (f\right )\right )} \sqrt{\frac{f}{\pi }}}{2 \, f}\right ) + \sqrt{2} \pi \sqrt{\frac{f}{\pi }} e^{\left (\frac{-i \, b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 i \, d f}{4 \, f}\right )} \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, f x + i \, b \log \left (f\right )\right )} \sqrt{\frac{f}{\pi }}}{2 \, f}\right ) - \sqrt{2} \pi \sqrt{\frac{f}{\pi }} e^{\left (\frac{i \, b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) + 4 i \, d f}{4 \, f}\right )} \operatorname{S}\left (-\frac{\sqrt{2}{\left (2 \, f x - i \, b \log \left (f\right )\right )} \sqrt{\frac{f}{\pi }}}{2 \, f}\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x} \sin{\left (d + f x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.28821, size = 405, normalized size = 2.85 \begin{align*} \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{8} \, \sqrt{2}{\left (4 \, x - \frac{\pi b \mathrm{sgn}\left (f\right ) - \pi b + 2 i \, b \log \left ({\left | f \right |}\right )}{f}\right )}{\left (-\frac{i \, f}{{\left | f \right |}} + 1\right )} \sqrt{{\left | f \right |}}\right ) e^{\left (\frac{i \, \pi ^{2} b^{2} \mathrm{sgn}\left (f\right )}{8 \, f} + \frac{\pi b^{2} \log \left ({\left | f \right |}\right ) \mathrm{sgn}\left (f\right )}{4 \, f} - \frac{i \, \pi ^{2} b^{2}}{8 \, f} - \frac{\pi b^{2} \log \left ({\left | f \right |}\right )}{4 \, f} + \frac{i \, b^{2} \log \left ({\left | f \right |}\right )^{2}}{4 \, f} - \frac{1}{2} i \, \pi a \mathrm{sgn}\left (f\right ) + \frac{1}{2} i \, \pi a + a \log \left ({\left | f \right |}\right ) + i \, d\right )}}{4 \,{\left (-\frac{i \, f}{{\left | f \right |}} + 1\right )} \sqrt{{\left | f \right |}}} - \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{8} \, \sqrt{2}{\left (4 \, x + \frac{\pi b \mathrm{sgn}\left (f\right ) - \pi b + 2 i \, b \log \left ({\left | f \right |}\right )}{f}\right )}{\left (\frac{i \, f}{{\left | f \right |}} + 1\right )} \sqrt{{\left | f \right |}}\right ) e^{\left (-\frac{i \, \pi ^{2} b^{2} \mathrm{sgn}\left (f\right )}{8 \, f} - \frac{\pi b^{2} \log \left ({\left | f \right |}\right ) \mathrm{sgn}\left (f\right )}{4 \, f} + \frac{i \, \pi ^{2} b^{2}}{8 \, f} + \frac{\pi b^{2} \log \left ({\left | f \right |}\right )}{4 \, f} - \frac{i \, b^{2} \log \left ({\left | f \right |}\right )^{2}}{4 \, f} - \frac{1}{2} i \, \pi a \mathrm{sgn}\left (f\right ) + \frac{1}{2} i \, \pi a + a \log \left ({\left | f \right |}\right ) - i \, d\right )}}{4 \,{\left (\frac{i \, f}{{\left | f \right |}} + 1\right )} \sqrt{{\left | f \right |}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]